ORDER AND CHAOS IN REVERSIBLE DYNAMICAL SYSTEMSbenji/PDF/roberts_physrep.pdf · 1. Introduction and...

CHAOS AND TIME-REVERSAL SYMMETRY.ORDER AND CHAOS IN REVERSIBLE

DYNAMICAL SYSTEMS

J.A.G. ROBERTS

Instituut voor TheoretischeFysica, Universiteitvan Amsterdam,Valckenierstraat65,1018 XE Amsterdam,The Netherlands

G.R.W. QUISPEL

Departmentof Mathematics,La Trobe University, Bundoora,Melbourne3083,Australia

NORTH-HOLLAND

PHYSICSREPORTS(Review Sectionof PhysicsLetters)216, Nos.2 & 3 (1992) 63—177. North-Holland PHYSICS REPORTS

Chaosand time-reversalsymmetry. Orderand chaosin reversibledynamicalsystems

JAG. Robertsa* and G.R.W. Quispe1’~°lnstituutvoor TheoretischeFysica, Universiteitvan Amsterdam,Valckenierstraat65, 1018XE Amsterdam,The Netherlands~ of Mathematics,La Trobe University, Bundoora, Melbourne 3083,Australia

ReceivedAugust 1991; editor: E.G.D. Cohen

Contents:

1. Introduction andoverview 66 4.1. Theory 1121.1. Time-reversalsymmetryand physics 66 4.2. Four classesof non measure-preservingreversible1.2. Time-reversalsymmetryandchaos 66 mappings 1161.3. Time-reversal symmetry and ordinary differential 4.3. Examples 122

equations 67 5. Conservativebehaviourin reversiblemappings 1261.4. Applications 70 5.1. KAM circles and their destruction 1271.5. Time-reversalsymmetry and mappings 73 5.2. Perioddoubling of symmetricfixed points 1371.6. Outline of this report 77 6. Dissipative behaviourin reversiblemappings 143

2. Review of somepropertiesof mappingsof the plane 79 6.1. Perioddoubling of asymmetricfixed points 1452.1. Definitions and properties 80 6.2. Strangeattractors 1522.2. Measurepreservation 86 7. Interplayof conservativeanddissipativefeaturesof revers-

3. Reversibility andmappingsof the plane 88 ible mappings;a symmetry-breakingbifurcation 1593.1. Some propertiesof reversiblemappings 89 8. Concludingremarksand future directions 1633.2. Generatingreversiblemappingsand involutions from AppendixA. Integrablereversiblemappings 166

symmetricsecondorderdifferenceequations 95 Appendix B. Nonlinear stability analysis for elliptic fixed3.3. Testing for reversibility and local reversibility 100 points 1693.4. Area-preservingmappingsthat arenot reversible 107 References 171

4. Non measure-preservingreversible mappings with asym-metric fixed points 111

Abstract:Dynamicalsystemswith independent(continuousordiscrete)timevariabletandphasespacevariablex arecalled reversibleif theyareinvariant

underthe combination { t—~— t, x—.* Gx) where G is sometransformationof phasespacewhich is an involution (Go G= Identity). Reversiblesystemsgeneraliseclassical mechanicalsystemspossessingtime-reversalsymmetry and are found in ordinary differential equations,partialdifferential equationsand diffeomorphisms(mappings)modellingmanyphysicalproblems.This report is anintroductionto someof thepropertiesof reversible systems,with particular emphasison reversiblemappingsof the plane which illustrate many of their basic features.Reversibledynamicalsystemsareshownto be similar to Hamiltoniansystemsbecausetheycan possessKAM tori, yet they aredifferent becausetheycan alsohaveattractorsand repellers.We createandstudyexamplesof thesehybrid dynamicalsystemsanddiscussthequestionof how to recognisewhetheragiven dynamicalsystemis reversible.

* Addressfrom July 1992: Departmentof Mathematics,Universityof Melbourne,Parkville, Victoria 3052, Australia.

0370-1573/92/$15.00© 1992 ElsevierSciencePublishersB.V. All rights reserved

I. A. G. Robertsand G.R. W. Quispel,Chaosand time-reversalsymmetry 65

__________________13:~ ~ ,~ i

~ :~..... ~~

— .__.._f á~—---... _________________________ Fig. 3.1. Fig. 1.3 with manyof thetrajectoriesremoved.The

j ~—“ X curved symmetry line of H = L * G. In colour are shownthe~ ,-‘ first threeiteratesof thesymmetry line of G under L, which

are the symmetry lines of L2°G, L4°Gand L6°G (cf.

3.14a). Theseare colouredgreen,red and blue, respectively.~ I’7 ~ _._ symmetryline y = .r of G is drawnin black togetherwith theAll symmetrylines passthroughthesymmetricfixed point atthe origin and the other symmetricfixed point at the otherintersectionof the lines of G and H. The intersectionof thesymmetry lines of L6 o G and G accountsfor theexistenceofthe symmetricsix-cycle. This cyclehastwo pointson eachof

___________________________________________________________ the symmetry lines of G and its iterates.The seven-cycle,

—13 beingodd, hasa point on thesymmetry line of G and of H.

~ 2135~

/ ,-_ —

Fig. 6.2. Illustration of thebehaviourof symmetrylinesin thevicinity of attractorsandrepellersin reversiblemappings.This

________________________________________________________ picture is fig. 4.lb, a phase portrait of example 1 when

C = 2.87, with theremovalof someof the orbits shownthereandthe addition of somesymmetrylines. The two innermostapparentlyclosedcurvesfrom fig. 4.lb are reproducedheretogetherwith one of the small islands surroundingthe sym-metric eight-cycle(right-handside).The symmetricfixed point

A at the centre of the picture is at the intersection of the(horizontal)symmetry line of G andthesymmetryline of H,

both of which are drawn in black. Two arrows mark the~.. .. attractingasymmetricfixed point(A) in thebottomhalf of the

picture andtherepellingasymmetricfixed point (R) in thetop. :.. -. . half. The colouredlines aresymmetrylines createdby iterat-

~ ~ ing thesymmetry line of G. The coloursdark blue, red and

- . -. - . greencorrespondto two, eight and sixteen iterationsof thisline under the mapping example 1 The colourslight blue

-. .. 0735 brown and purple correspondto 2, 8 and 16 iterationsof the12065 2-2065 line under the inverseof example1.

1. Introductionand overview

This report is a review of someof the propertiesof dynamical systemspossessingtime-reversalsymmetry. In this report asystemthat is invariantundertime reversalwill be called reversible*~.Theclassicalconceptof time-reversalsymmetryrefersto the invarianceof equationsunderthe transforma-tion t—~— t (this definition will be generalisedbelow). In this introductorychapter,we briefly discussthe place that time-reversalsymmetry occupiesin physics and look at the consequencesthat thepossessionof time-reversalsymmetryhas for nonlinearclassicaldynamical systems(cf. also Quispel[1992a]).We then describereversiblemappings,which havetime-reversalsymmetry where time isdiscrete.Our review will focuson suchmappings.

Li. Time-reversalsymmetryandphysics

Time-reversalsymmetryhasplayed,andstill plays, an importantrole in physics(cf. Davies [1974],Sachs[1987],Thom [1980,1990], Zocher and Török [1953]).Many of the differential equationsofphysics are time-reversible. This was first noticed by Loschmidt 11877] for particles moving in avelocity-independentforce field. Boltzmannsoonrealizedthe importanceof time-reversalsymmetryand later showed that Maxwell’s equationsare~reversible (if one also reversesthe field B—* — B)[Boltzmann1897 a, b, 1898]. Painlevé[1904]presentedan earlyuseof the time-reversalinvarianceofNewton’sequationsof motion for a free-falling body (cf. also Sachs[1987,chapter2]). The Einsteinequationsof classicalgeneralrelativity arereversible[Penrose1979, 1989],asaretheequationsof, e.g.,the famousthree-bodyproblem (cf. Marchal [1990]).Finally, Wigner hasshown the importanceoftime-reversalsymmetryin quantummechanics[Wigner1959; Doncel 1987].

In spite of the fact thatphysical lawsmay be invariant undertime reversal,physicalphenomenaingeneralarenot. This long-recognised(Loschmidt’s)paradoxis discussedin Penrose[1979]:“The localphysicallaws we know andunderstandare all symmetricalin time. Yet on a macroscopiclevel, timeasymmetryis manifest.In fact, a numberof apparentlydifferentsuchmacroscopicarrowsof time maybe perceived.”Penrosegoeson to list sevenapparentlyindependentarrowsof time asdefinedby thedecayof the K°-meson,quantummechanicalobservations,entropyincrease,retardationof radiation,psychologicaltime, expansionof the universe,and black holes versuswhite holes (cf. also Davies[1974],Hawking[1987,1988],CoveneyandHighfield [1990],Page[1992]).To theseonemight perhapsadd thearrowdefinedby biological evolution. The usualexplanationof theabove-mentionedparadoxis that althoughthe lawsof physicsareinvariantundertime reversal,the initial conditionsaresuchthatthe time evolution is not [Sachs1987; Penrose1989].

1.2. Time-reversalsymmetryand chaos

In recentyears“chaos” or, more properly, nonlineardynamicshasbecomean importantareaofresearchactivity. A traditional division of classicalnonlineardynamicalsystemsis into the classof

*) Note that dynamic reversibility, asdefinedhere, is not necessarilyequivalentto invertibility or to thermodynamicreversibility.

66

J.A.G. Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry 67

conservativesystemsand the class of dissipative systems. By “conservative” systemswe meanHamiltonianordinarydifferentialequationsandsymplecticmappings.Dissipativesystems,on the otherhand,are characterisedby collapseof their motion after some transienttime onto an attractor(formoredetails aboutconservativesystemsand dissipativesystems,seefor exampleMacKay and Meiss[1987],and Cvitanovic [1989],respectively).

In comparison,reversibledynamicalsystemshavereceivedfar lessattentionand the treatmentthattheyhavereceivedhastendedto beless systematic.This hasled to thesituationwherethe literatureonthe subjectis quitescattered,with someauthorsnot beingawareof the generalityof the mathematicaltheoryunderlyingtheir results. Oneof our motivationsfor writing this report is to try andremedythissituation.For the samereasonwe providean extensivelist of referenceson (nonconservative)reversiblesystems,both on their theory, and on their applications in such diverse fields as condensedmatterphysics,fluid dynamics,laserphysics, moleculardynamics,quasicrystals,chemistry, biology etc. (formorereferenceson reversibledifferential equations,see Sevryuk [1991b]).

Although this reportconcernsclassical dynamicalsystems,we mention that reversibility plays animportant role in “quantum chaology”, i.e., the studyof semiclassical,but nonclassical,behaviourofsystemswhose classical motion exhibits chaos [Berry 1987]. In such quantumsystems,reversibilitymanifestsitself in the statisticsof the spacingof energylevels (cf. Berry [1987],Ozorio de Almeida[1988],Keating[1990]).Thesestatisticsfall into universalityclasses.If the underlyingclassicalsystemisintegrable,the statisticsof level spacingsis that of a setof randomnumbers(i.e., Poissonian).If theclassicalsystemis chaoticand reversible, the level spacingsobey the eigenvaluestatisticsof infiniterandomreal symmetricmatrices.If theclassicalsystemis chaoticandnot reversible,the spacingshavethe samestatisticsas the spectraof randomcomplexhermitianmatrices.Thus time-reversalsymmetrybreakinginducesa spectralphasetransition (cf. Lenzand Haake[1990]).

1.3. Time-reversalsymmetryand ordinary differentialequations

Let usnow bemathematicallyabit morepreciseas regardsa definitionof reversibility.Traditionally,what springsto mindwhenonethinksof time-reversalsymmetryis a systemdescribedby an equationof the form

x=F(x), F:l~—*O~ (1.1)

(herethe dot denotesdldt). This equationdescribes,for example, the motion of a particlewith unitmass moving along a line under the influence of a conservativeforce F = F(x). This equationisreversiblebecauseit is invariant undertime reversal:t —* — t (andhenced/dt—~ —dI dt). An immediateconsequenceof the reversibility of (1.1) is that if x = y(t) is a solutionof the equation,thenso is thetime-reversedmotion y(— t). The lattersolution is what would be seenif a movie was madeof y(t) andthenplayed backwards.

It is easyto seethat the following equationsarealso invariant under t —* —

i=F(x,x2), F:l~2—~lJ~ (1.2a)

x=F(x), F:IRm—sl~m, (1.2b)

for arbitrary(analytic) functionsF. Thereis, however,onebig differencebetweenequation(1.1) on the

68 JAG. Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry

one hand and (1.2a,b) on the other. Equation (1.1) is conservative— it is derivable from theHamiltonian

H(x, p) = ~p2+ V(x), (1.3)

with p = I (recall mass= 1) and F(x) = —dVldx, and the energyE = H is conserved.Equations(1.2a,b) arein generalnot conservative.

This leadsusto the main themeof thisreport: the similaritiesanddifferencesbetweenreversibleandconservativesystems.From the above examples,it is clear that thereexist systemsthat are bothreversibleand conservative,and othersthat arereversiblebut not conservative.The actual situationisindicatedin fig. 1.1. RegionII representssystemsthat are both conservativeand reversible.Thesesystemsare thosewherehistorically reversibility was first noticed and region II is the most studiedregion in the figure. It includessystemsof the form (1.1), or equivalently (1.3), and more generallyincludesanysystemdescribedby a Hamiltonian which is evenin the momenta,i.e.,

H(x,p)=H(x,—p), x,pEIlr. (1.4)

Studiesof Hamiltonian systemslike thesein which the time-reversalsymmetry hasbeenspecificallystudiedor utilised in the analysisincludeinvestigationsby de Vogelaere[1958],Meyer [1981],vanderSchaft [1983],deAguiar et al. [1987],de Aguiar andMalta [1988],Ozorio de Almeidaandde Aguiar[1990],andMontaldi etal. [1990a,b]. RegionI of fig. 1.1representssystemsthat arereversiblebut notconservative.This includessystemsof the form (1 .2a,b). Finally, region III comprisingsystemsthatareconservativebut not reversibleis not empty, althoughexamplesare not as easyto find. An examplegiven by Sevryuk [1992]is the systemwith Hamiltonian

H(x, p) = H (a1x + b1p). (1.5)

For generalparametersa, andb~,the flow generatedby this Hamiltonianis not reversible.

C

III

R I II

Fig. 1.1. Venn diagram showingthe relationbetweenreversibledynamicalsystems(setR) andconservativedynamicalsystems(setC).

J.A. G. Robertsand G.R.W. Quispel, Chaosand time-reversalsymmetry 69

The invarianceunder time reversal:t—~—t of (1.1) shows up as invarianceunder t—~—t, x—~x,p—~ —p of the equivalenttwo first-orderequationsderivablefrom the Hamiltonian(1.3).Similarly, theequationsof motion obtainedfrom Hamiltonianswith the property (1.4) are invariant undert —~ —

x —~ x, p—~ —p. Although not generally Hamiltonian, eq. (1 .2a) can of coursealso be rewrittenequivalentlyas a systemof coupledfirst-order equations,

£=y, ji=F(x,y2), (1.6)

andthe time-reversalinvariancein this descriptionmanifestsitself in a similar way to the Hamiltoniancase.That is, eqs. (1.6) areinvariant under t—~—t, x—~x, y—~—y; now in the two-dimensionalphasespace,if (x(t), y(t)) is a solutionof the dynamics,thenso is (x(— t), —y(— t)). The lattercorrespondstoflipping the trajectory (x(t), y(t)) via the transformationof phasespacethat leaves the coordinateunchangedand changesthe sign of the velocity, and following the resultingtrajectory in the oppositetime sense.The drawbackwith this way of defining reversibility in phasespaceas invarianceunder

— t accompaniedby reversalof velocity y —~ —y (or momentump—~ —p) is that this definitionis notcoordinate-invariant.It is alsotoo restrictive.The following definition remediesthesedefects[Devaney1976]:

A dynamicalsystem(notnecessarilyconservative)is reversibleif there is an involution in phasespacewhich reversesthe direction of time.

This definition appliesboth to ordinary differential equationsand also to mappings(seesection1.5).An “involution”, which we will oftendenoteby G, is a transformationthat composedwith itself yieldsthe identity (Go G = Id).

Thus, a generalsystemof first-ordercoupledordinarydifferential equations,

dxldt=F(x), xEIlr, (1.7)

is reversibleif thereis an involution G which reversesthe direction of time, i.e.,

d(Gx)/dt= —F(Gx), (1.8)

andhence

dG.F—FoG. (1.9)

HereF andG arefunctionsfrom Il” to itself. The abovedefinitionmeansthatunderthetransformationof the n-dimensionalphasespaceby G, the system(1.7) transformsto that obtainedby just putting

— t, so that under the combinedaction of the involution and time reversalthe equationsareinvariant. An involution that achievesthis is often called a (reversing) symmetry of the system.Trajectoriesthat are left invariant by G are calledsymmetric;otherwisethey are asymmetric.In theaboveexamples(1.4) and(1.6) the involutionsareG: x—~x,p---~—p andG: x—~x,y—~—y, respective-ly. Note that this generaliseddefinition of reversibility is no longer restrictedto even-dimensionalsystems.Moreoverin even2m-dimensionalsystems,the setof fixed pointsof the involution G, whichplaysan important role in the dynamics,neednot havedimensionm.


Althoughatfirst sight it maynot seemclear, this newdefinition of reversibility encapsulatesexactlywhatoneintuitively thinksof astime-reversalinvariance— a consequenceof it is that if x(t) is asolutionof (1.7) thenso is Gx(— t), which correspondsto a reflection of the trajectoryx(t) in the n-dimensionalphasespacefollowed in the oppositesense.Note that whenthe involution Gis linear, the conditionforthe system(1.7) to be reversible [Moser1973] is that G changesthe sign of the vector field F, i.e.,

GF(x) = — F(Gx). (1.10)

Reversible differential equations in which G is linear, feature prominently in physically relevantexamplesin the literature (seesection 1.4 andreferencestherein).Thereasonwe allow in the generaltheory for an arbitrary (nonlinear) involution G is that, althoughsimple linear involutions such asG: x—~x,y—~—y can becomevery complicatedunder a coordinatetransformation, they remaininvolutions. Being an involution is a coordinate-independentproperty.

Apart from generalisingthe definition, Devaney[1976]went on to show that thereare very closesimilarities betweenthe symmetricperiodic orbits of a (generalised)reversibleflow and the periodicorbitsof a Hamiltonian flow (in particularsymmetricperiodicorbits in a largeclassof reversibleflowscomein one-parameterfamilies, just as periodic orbits in Hamiltonianflows), cf. alsoDevaney[1977].Earlier,Moser [1967,1973] hadshownthat families of quasiperiodicmotionsor Kolmogorov—Arnol’d—Moser (KAM) tori, previouslyassociatedwith Hamiltonian systems(cf. Amol’d [1961],Moser [1968],Siegel andMoser [1971,section36]), exist also in reversiblesystems,cf. alsoBibikov [1979],Scheurle[1987],bossand Los [1990]and referencestherein. In particular, such KAM tori are found in thevicinity of linearly stablesymmetric equilibrium points or symmetricperiodic orbits. More recently,thesesimilarities havebeenfurther emphasisedby Arnol’d [1984],Arnol’d and Sevryuk [1986],andSevryuk [1986,1987,1990,1991a],who haveextendedsomeof the KAM resultsfor generalreversibledifferentialequations.Furthercontributionstowardsestablishingparallelsbetweenreversiblesystemsand Hamiltonian systemsinclude those by Vanderbauwhede[1986,1990,1992], Palmer [1977]andothers(seethe comprehensivebibliographiesin Sevryuk[1991a,b]).

This is not to saythat reversiblesystemshaveto be Hamiltonian.The above(generalised)definitionof reversible systemsexplicitly includes systemsthat are not Hamiltonian (region I of fig. 1.1).Reversiblesystemsin this region can haveasymmetricperiodicorbits that areattractors.Becauseof thetime-reversalsymmetry,the presenceof anattractorA implies thepresenceof a repellerR = GA givenby the reflectionof the attractorby the symmetry.This is illustrated by theso-calledlogisticdifferentialequationx = x(1 — x). This equationis invariant under 1—3 —t, combinedwith the (one-dimensional)phasespaceinvolution x—> 1 — x. It is easyto verify that the fixed pointx(t) 1 is attracting,andthatits reflection x(t) 0 is repelling.

More generally, nonconservativereversible differential equationscan display Hamiltonian-likebehaviourneartheir symmetriccyclesas well as behaviourtypical of dissipativeandexpansivesystemsneartheir asymmetriccycles. This hybrid nature makesthem interestingmodelsin which to studywithin a single systemthe differencesand crossoverbetweenconservativeand dissipativebehaviour.

1.4. Applications

To illustrate the range of problemsthat can be described by reversiblesystemsthat are notHamiltonian, in this sectionwe give somephysically motivated examplesof reversibledifferentialequationsbelongingto region I of fig. 1.1.

JAG. Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry 71

(i) An externally injected class B laser (cf. Arecchi [1987])can be describedby three couplednonlinearordinary differential equations.When the damping rate associatedwith the field is muchgreaterthanthatof the population,as in CO2 lasers,Politi et al. [1985,1986a,b] haveshownthat thissystemof equationsis

I=zx+y+C1, y=zy—x, i=C2—x2—y2. (1.11)

Here C1 and C2 are parameters. The system (1.11) is reversible with respect to

G: (x, y, z)—1. (—x, y, —z). However, the divergenceof the vectorfield on the right is 2z, illustratingthat phase-spacevolume is not conserved.For realistic values of the physical parameterstheseequationswere studiedby Politi et al. They observedthe coexistenceof “dissipative-like structures”,i.e., periodic attractorsandrepellers,as well as KAM tori. Moreover the attractorsand repellersarosefrom a symmetricperiodic orbit via a symmetry-breakingbifurcation.

(ii) In nonequilibriummoleculardynamicssimulationsof a fluid or solid, Newton’s equationsofmotion are integratednumerically and thermodynamicquantitiesare relatedto long-time dynamicalaverages.Often the equationsof motion areaugmentedby addingadditionalvariablesso as to achievethe constancyof somethermodynamicquantitieslike temperatureor pressurethroughoutthe dynamics,as canbe realizedexperimentally.The resulting“extended”systemof equationsis often not derivablefrom a Hamiltonian, althoughstill time-reversible[Evans1986]. An exampleis provided by Nose—Hoover dynamics[Nose1984, 1986; Hoover1985, 1986, 1988] in which the additional variablesaretime-dependent“thermodynamicfriction coefficients” which act so as to keep the averagekineticenergy(temperature)constantwhilst yielding the canonicalensemblein the “unextended”spaceofphysicalvariables(cf. alsoJellinek [1988],Jellinek and Berry [1988,1989], Hoover[1989],andBulgacandKusnezov[1990]).A simple illustration of this Nose—Hooverdynamicsis provided by a particlewith unit mass moving along a line underthe influenceof a periodic potentials(1 — cosx) and fixedexternal force F [Hoover et al. 1987]. The particle has coordinatex and momentum y and itstime-averagedkineticenergyis keptconstantvia a feedbackmechanisminvolving afriction coefficientz. The equationsof motion are

x=y, .~=F—rsinx—zy, i=a(y2—1), (1.12)

wherethe parametera>0 is called the thermostatstrength.The friction coefficientz works so as toreducethe magnitudeof the momentumy if the kinetic energyy2/2 exceedsthe equilibrium value1/2.On the otherhand, if the kinetic energyis lessthan 1/2, i decreasesso that z can becomenegative,leading to an increase in the magnitude of y. Equations (1.12) are reversible with symmetryG: (x, y,z)—~(x, —y, —z). Numerical investigationsof theseequationshaverevealedthe presenceofattractinglimit cycles and fractal chaotic attractorsin the 3D phasespace.Numerical simulationsofanalogousreversibleequationsof motion for systemswith severalcoupledoscillatorshavealsorevealedcharacteristicdissipativebehaviourin the form of the phasespacevolume shrinkingdown to astrangeattractorassociatedwith steady-statetrajectories(cf. Holian et al. [1987],Hoover[1988]andreferencestherein).

(iii) The sedimentationof small spheresin a fluid, in the limit of infinite viscosity,hasbeenstudiedby Hocking [1964],Caflisch et al. [1988]and Golubitsky et al. [1991].Approximatingthe spheresbypoint particles, the equationsof motion for n particlesmoving jn a polygonal configuration are


i1,2,...,n; U(r)=ft+~1~~

(1.13)

wherethe indices are takenmodn. Herer1,. . . , r, E ~ denotethe consecutiveedgesof an n-sidedpolygon in ~, ande~is a unit vector in the z direction (in which gravity is acting). Equation(1.13) isreversiblewith symmetry G: (x1, y1, z1, x2, y2, z2, . . . , x,,, y0, z~)—÷(—x1,y1, z1, —x2, y~,z2,. .

—x,,,y,,, Zn); this reversibility can be used to prove the existenceof periodic and quasiperiodicsolutionsfor themotion. [Notethat the divergenceof theright handsideof (1.13) is zero,i.e., (1.13) isvolume-preserving,andhencepossessesno attractors.]

(iv) Recently,Tsang et a!. [1991a,b] have studiedthe following systemof N coupledordinarydifferential equationsmodelling electrical circuits comprisingseriesarraysof Josephsonjunctions inparallelwith a single resistor,

~=i+acosO+~~cos~. (1.14)

Herek = 1, 2,... ,N,0k arephaseangles,definedmodulo21T, and£2 anda areparameters.Equations

(1.14) areinvariant under°k — 0k with t —~ — t. For N= 2, numericalinvestigationshaveshown,asin(i) above,coexistenceof Hamiltonian-likebehaviourwith dissipationandexpansionvia thepresenceofa sink (attractor) and source (repeller). In other recent papers,Goncharet al. [1991]study theinteraction in reversiblehydrodynamicequationsbetweena sink—sourcepair of point-like vorticesanda potentialwave,andKent andElgin [1991a,b] considera reversiblesystemof threecoupledordinarydifferential equationsrelevant to travelling wave solutions of the Kuramoto—Sivashinskyequation,explaining someobservedbifurcation behaviourof the periodic orbits of this systemin terms of thereversibility.

We should remark at this point that in many applicationsoneencountersreversiblesystemswherethe independentvariableis a spatialvariable,ratherthana temporalone(e.g. MacKay [1987],BaesensandMacKay [1991]).In termsof thephysical,or chemical,or biological interpretationthismaymakeadifference,but mathematically,of course,the two symmetriesarecompletelyequivalent.For the sakeof conveniencewe will alwaysusethe languageof time reversal.Examplesfrequently arisein partialdifferential equations,and in partial differential—differenceequations,when the stationarysolutionssatisfy reversible ordinary differential equations (e.g. Turing [1952], Eckmann and Procaccia[1991a,b]).

(v) The evolutionof a gasflamefront in the presenceof anexternalstabilizingfactoris describedbya partial differential equationbelonging to the classof equationsof the form

+ ~ 2a~+~+ + ~ +

where ~= ~(x, t) is the displacementof a flame point from its unperturbedposition, x is the spatialcoordinate,t is time anda, f3, ‘y, and6 areparameters[MalomedandTribelsky 1984]. Thestationarysolutionsof this equation~ = ~(x) are given by

(1.15)


which also describesthe stationarysolutionsof nonsteadylaserevaporationof condensedmatter.In thefour-dimensionalphasespaceof variables(~, ~, ~ ~ eq. (1.15)can be rewrittenas a first ordersystemof four coupledordinarydifferential equations(in the usual way) and is invariant under thecombinedapplicationof the involution G: (~, ~, ~, ~ —3 (~, — ~, ~, — ~ andx —~ — x. Thus itis reversiblewith x playing the role of “time”. This reversibility accountsfor the existenceof aone-parameterfamily of periodic solutionsof (1.15) andsomerelatedstability resultsof thesesolutions[Arnol’d 1984; Arno!’d and Sevryuk 1986; Sevryuk 1986; Sevryuk 1989].

(vi) Other reversiblepartial differential equationshavebeenstudiedby Kirchgässner[1982,1983],Renardy[1982],Fieldet a!. [1991],andvanSaarloosandHohenberg[1991].The equationsstudiedbyKirchgassnerinclude onesthat mode! permanentwavesin density-stratifiedchannelsand viscousfluidflow betweenconcentriccylinders. The equationsstudiedby Renardyhaveapplicationsto reaction—diffusion equations,describing, e.g., the Be!ousov—Zhabotinskiireaction. Vanderbauwhede[1986,1990,1992] hasproved theoremsshowing the prevalenceof bifurcationsof subharmonicsolutionsinreversibleordinary differential equationswhich include the Mathieu equation.Someof theseresultshaverecentlybeenillustratedin a computationalstudy of a systemof ordinarydifferential equationsdescribingsteady-statesolutionsof a systemof two reaction—diffusionequations[KazarinoffandYan1991].

1.5. Time-reversalsymmetryand mappings

We turn our attention now to mappings(diffeomorphisms),on which we will concentratein thisreport. In this sectionwe will introducereversiblemappings,i.e., reversibledynamicalsystemswithdiscretetime. Recall the definition of reversibility givenin section1.3.A dynamicalsystemis reversibleif there is an involution in phasespacewhich reversesthe direction of time.

Considera mappingL that associatespairs of pointsof somespace(e.g., B~)via

x,~1=Lx~,iEl. (1.16)

The set of points {x1, x2,x~,.. .} formed by repeatedapplicationof the mappingto the point x0constitutesits orbit, a discreteanalogueof the continuoustime trajectory of an initial conditionof adifferential equation.The mappingL is reversibleif thereexists an involution G such that

LoGx~~1=Gx,, (1.17)

where o denotesthe compositionof two mappings.Whereasin eq. (1.16) we havethat x,~1is thesuccessorof x, underL, eq. (1.17) showsthat, underthe transformationof phasespaceby G, we find

= Gx~~1is now the predecessorof y, = Gx1. That is, time evolution is reversed.From (1.17)and(1.16),

~ (1.18)

Sincethis equationholds for arbitrary x., we get the defining property of a reversiblemapping

LOGOL=G, G0G=Id, (1.19a,b)

whereId is the identity mapping. G is called a (reversing)symmetry of L.

74 J.A.G.Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry

A reversib!emappingcan be written as the product (i.e., composition)of two involutions

L=H0G, GoG=H0H=Id. (1.20a,b)

This follows from (1.19a,b) with H:= LoG since

L=(L0G)oG=H0G, HOH=(LOG)o(LoG)=Id. (1.21a,b)

Conversely,anymappingthat decomposesas a productof two involutionsU andR, i.e., L = UoR, isreversiblewith, e.g.,G = U or G = R, so that (1.20a,b) is an equivalentway of defining a reversiblemapping.One applicationof a reversiblemappingthen amountsto applyingtwo reflections,oneafteranother.

An important featureto note is that the definition of reversibility for mappings(1 .19a,b) or thealternative definition (1.20a,b) is completelyalgebraicand does not rely on any differentiabilitypropertiesof the mapping. This is the reasonone can, for example,speakof cellular automatawithtime-reversalsymmetry [Vichniac1984; Margo!us 1984; Pomeau1984; Takesue1987, 1989, 1990a,b;Toffoli and Margo!us1990; Ab!owitz et al. 1991] (andreferencestherein).

The major classesinto which dynamical systemswith continuoustime are divided, and theirdistinguishing features,have close analoguesin mappings.Moreover, the study of mappingshasnumericaland some conceptualadvantagesover the study of differential equations.It is now we!!recognisedthat evena two-dimensionalmapping,a mappingof the plane,containsa wealthof exoticdynamicalbehaviour(cf. He!leman[1980],Hénon [1983],LichtenbergandLieberman[1983]).There-fore considernow the Venn diagramof fig. 1.1 in the context of mappings.

Hamiltonian ordinarydifferential equationshaveas their mappinganaloguesymplecticmappings.Both of thesetypes of systemsarecombinedin fig. 1.1 under the heading“conservative” dynamicalsystems.In two dimensionsthe only requirementfor a map to be symp!ectic is that it is volume-preserving,i.e., area-preserving.Area-preservingmappingsare widely discussedin the literature (cf.MacKay [1982],Hénon[1983],MacKayandMeiss [1987]).They can beobtainedfrom the “surfaceofsection” of a four-dimensionalHamiltoniansystemwith motion on a three-dimensionalenergysurface(cf. Poincaré[1912], Hénon [1983],Lichtenberg and Lieberman [1983]). The possible motions inarea-preservingmappingsinclude periodicorbits, orbits that escapeto infinity and,significantly, orbitsthat fill continuous curvesin the plane. There is a KAM theoremthat holds for area-preservingmappings[Moser1973]. Under certainconditionsit guaranteesthe presenceof infinitely manyclosedinvariantcurvesin the planeon whichthe motion is quasiperiodic.Betweenthesecurvesarethin bandsof chaoticorbits.Note that in higher dimensions,D >2, volume preservationis no longer sufficient tomake a mapping symp!ectic so that symp!ectic mappingsare a strict subsetof volume-preservingmappings.

A reversiblemappingis an analogueof a reversibledifferential equation.The defining properties(1 .19a,b) or (i.20a, b) given aboveguaranteethat the applicationof a (reversing)symmetry G to anorbit of the mappinggives anotherorbit of the mappingwhen followed in the oppositesense,theessenceof (continuous)time reversalas discussedin section1.3 (cf. fig. 1.2 for anillustrationof this intwo dimensions).The “surfaceof section” mappingof a reversibledifferentialequationis a reversib!emapping (cf. de Voge!aere[1958])and historically this is how they arose (cf. the study of Birkhoff[19151of the restrictedthree-bodyproblem,a Hamiltonianproblem).The time-i mapof a reversibledifferential equationis alsoa reversiblemapping. Reversiblemappingscan alsoarisefrom differential

l.A. G. Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry 75

~ 2(OG~3

Fig. 1.2. Illustration of the natureof the motion in a reversiblemapping L of the plane with symmetry G: x-+y, y— x. Shownis part of thetrajectoryof apoint x0 with unbrokenarrowsindicating the actionof L on eachpoint. Reversibility implies thatthetrajectoryof Gx, is foundbyreflectingthetrajectoryof x, by G. The forward and backwardtrajectoriesare interchangedon reflection as the brokenarrowsindicate.

equations in less obvious ways, e.g., in the theory of arbitrary (i.e., not necessarilyreversible)discontinuousordinarydifferentialequations(cf. Teixeira[1981,1990]). In suchequations,which arisein controltheory,economicsandnonlinearoscillations,the vectorfield on the right-handsideof (1.7) isdifferent in two different parts of phase space separatedby a surface of discontinuity. In theneighbourhoodof points on this surface,the surface-of-sectionmapping can be describedas thecompositionof apair of involutions.

Surface-of-sectionmappingsand time-i maps cannot usually be calculatedexplicitly from anunderlying differential equation (a numerical integrationschemefor reversibleordinary differentialequations,which preservestime-reversalsymmetry, is presentedin Quispe! [1992b]). Nevertheless,explicit mappingscanbe deriveddirectly for variousproblemswithout the needto numericallyintegrateadifferentialequation.In particular,explicit reversiblemappingscan beanalytically derivedfrom somekicked reversibleHami!toniansystems(i.e., systemssubjectedto a regularly applied impulsive force)that occur as models in solid statephysics (cf. Heagy and Yuan [1990]andreferencestherein) andpartic!eacceleratordynamics(cf. Helleman[1979,1980]). Explicit reversiblemappingsalsofollow fromproblemsmodelledby symmetricdifferenceequations,e.g.,

x,~1 + x._1 f(x,); (i.22a)

~ g(x,y)=g(y,x), (1.22b)

cf. section3.2. Thesedifferenceequationsaresymmetricbecausetheypropagatevia the sameformulain forward and backward time. Equations (1.22a,b) with appropriateidentifications lead to two-dimensionaland three-dimensiona!reversiblemappings,e.g., respectively,L: x1 = (x,_1,x,) —* x,~1=

76 JAG. Robertsand G.R.W.Quispel,Chaos and time-reversalsymmetry

(x~,x,÷1) with symmetry ~ and ~(x,, x~+1,x~÷2)with symmetry G: 1, = (x~_1,x,, x,+1)—+x~~1=(x,~1,x,, x,_1). Symmetric differenceequations such as these (and hence reversible mappings) arise in, for example, models of the propertiesof one-dimensional crystals and quasicrystals [Janssenand Tjon 1983; Kohmoto 1987; Baake et a!.1992; Baake and Roberts 1992], spin chains [Belobrovet a!. 1984; Roberts and Thompson 1988] andparticle chains [Aubry 1983; Johannesson et al. 1988], and from looking at stationary statesofdifferential—differenceequations[Quispel et a!. 1988, 1989; Lahiri and Ghosa! 1987, 1988], andtravelling wave solutions of partial differencesoliton equations[Quispe!et al. 1991, and referencestherein].

We now give examplesof mappingsin eachof the threeregionsof fig. 1.1. The moststudiedsystemsare (again) thosethat are both reversible and symplectic (region II in fig. 1.1) see,e.g., MacKay[1983a,1986], and DewarandMeiss [1992]for area-preservingmappingsof the planeandHu andMao[1987] and Kook and Meiss [1989] for higher-dimensionalsymplectic mappings (for studies ofhigher-dimensionalreversiblevolume-preservingmappings,cf. de Vogelaere[1958],Hu andMao [1987]and Mao [1988],and referencestherein). In two dimensions,a we!! known examplefrom region II isthe area-preservingHénonmapping[Hénon 1969],

x’=y, y’=—x+2Cy+2y2 (1.23)

[note that we departfrom the subscriptnotation of (1.16) and hereafteruseunprimedand primedvariablesfor a point (x, y) andits image(x’, y’)]. This reversiblemappingfollows from the symmetricdifferenceequation(1 .22a)with f(x~)= 2Cx

1 + 2x~andthe identificationabove. It can be written as thecompositionof two invo!utions Ho G, with

= -y + 2~x+ 2x2, G:{~:= Y~ (1.24)

Another example of a reversible area-preservingmapping is the Chirikov—Tay!or or standardmapping(cf. Taylor [1969],Froeschlé[1970],Chirikov [1979],Greene[1979]),

x’=x+y, y’=y+(K/21T)sin2lTx’, (1.25)

which arises from models of simple pendu!a and accelerator dynamics [Chirikov 1979, 1987] and groundstates of the Frenkel—Kontorova mode! [Aubry 1983], and has been widely studied*). It follows from(i.22a) with f(x1) = 2x1 + (K/2ir) sin 2irx1 and the identification (x,, x,~1 — x,)*-+(x, y), (x~~1,x,.~2—x~~1).E-~(x’,y’) [note that in this report we will often use— as in (1.25)—the compact notation ofwriting the primedvariableof oneof the definingmappingequationsin the right-handside of the otherdefining equation].The standardmappinghasthe decompositioninto involutionsHo G with

•fx’=x, fx’=x+y, 126H~11yi= —y+ (K!21T) sin2ITX, ~ = —y.

Oneof the practicalbenefitsof studyingarea-preservingmappingsthatare alsoreversibleis that inreversiblesystemsmanyof the periodic orbits can be found by searchingfor a point of the orbit along

~ Note that eq. (1.25) looks slightly different to what mostauthorscall thestandardmappingbut is in fact trivially relatedto it.

I.A.G. Robertsand G.R.W. Quispel,Chaos and time-reversalsymmetry 77

the fixed !inesof the componentinvolutions.Usually reversingsymmetriespossesssuch a line of fixedpoints [e.g., y = x in the caseof G in eq. (1.24)]. The periodic orbits found in this way are thesymmetricperiodic orbits.

As for differential equations,the reversibility property defined for mappingsis independentofwhetherthe mappingis conservative.Indeed,in agenera!reversiblemappingpartsof the phasespacemay be contractedwhilst other parts may be expandedduring the motion. It is only recently thatnonconservativereversiblemappings(regionI of fig. 1.1) havereceivedattention.Arno!’d andSevryukhaveproved KAM theoremsfor reversiblemappingsthat need not be conservative[Arnol’d 1984;Arnol’d andSevryuk1986; Sevryuk1986,1990] (cf. alsoSevryukandLahiri [1991]and RoyandLahiri[1991]for resultson Hopf-typebifurcationsin four-dimensionalreversiblemappings,as well as Lahiri[1992]). In two dimensions the reversible mapping KAM theoremsguaranteethat under certainconditionsthe neighbourhoodof a symmetricperiodic orbit bears“striking similarities” [Arnol’d 1984]to the neighbourhoodof a periodic orbit in an area-preservingmapping.That is, it containsanestedsequenceof invariant curves,islandchainsand chaoticbands.A reversiblemappingwith bothKAMcurvesand periodic attractorsand repellerswas discussedby Politi et a!. [1985,1986a,b]. It was notgiven explicitly but instead arose from taking a surfaceof section of the reversiblesystemof threecoupledordinarydifferentialequationslistedin eq. (1.11) above.Bul!ett and coworkers[Bu!lett et al.1986; Bullett 1988, 1991] havestudieda class of reversiblemappingsin the complexplanecontainingattractingand repelling periodic orbits (cf. also Meste! andOsbaldestin[1989]).

An exampleof a reversiblenon area-preservingmapping(region I of fig. 1.1) is

(C — y)[l + (y’ — 1)2], y’ = x/[1 + (C — y — 1)2]. (1.27)

This mappingis composedof the two involutions(cf. chapter4),

H.IxE1+~_1fl, ~jXX, 128

~~yl=x/[i±(y_1)2], ~y’=C—y. ( . )

An examp!eof an area-preservingnonreversiblemapping (region III in fig. 1.1) is (cf. section3.4),

x’ x—y2(1—y), y’ =y+ Cx’2(i—x’), C~0,Cst1. (1.29)

It will be shown in chapters3 and4 how onecanconstructboth reversibleandnonreversiblemappings.Finally we note that thereis no test for global reversibility. However,necessaryconditionsfor a localform of reversibility are discussed in section 3.3.

1.6. Outline of this report

In this reportwe will studytime-reversalsymmetryin classicaldynamicalsystemsvia a surveyof thedynamicsof reversiblemappingsof the plane.The reasonfor this restrictionis that two-dimensionalmappingsare, in a sense,the simplestreversiblesystemscapab!eof exhibiting bothchaoticbehaviourandorderedbehaviourin different regionsof phasespace.The fact thatarea-preservingmappingsofthe plane possesssome canonical featuresof conservativesystemsalso a!lows us, at the level oftwo-dimensionalmappings,to comparethe relationshipbetweenconservativeand reversiblesystemsasdepictedin fig. 1.1 (cf. also Robertset al. [1991]).

78 J.A.G. Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry

We will to a largeextentconcentrateon nonconservativereversiblesystems(regionI of fig. 1.1).Theinvestigationof region I of fig. 1.1 is facilitated by the use of systematicmethodsfor creating!argeclassesof reversiblenonconservativemappingswith both attractorsand repellers(cf. Quispel andRoberts [1988,1989], Roberts [1990a,b]). The explicit reversiblemappingsthus obtainedare mucheasierto studythanonesdefinedby a surfaceof section.They graphicallyillustrate the fact that thephaseportrait of a genera!reversiblemappingtypically containsboth KAM tori, the distinguishingfeature of area-preservingmappings,as well as attractors, the distinguishingfeature of dissipativemappings(cf. fig. 1.3). The fact that moststudiesof reversiblemappingsto datehavebeenrestrictedtoarea-preservingreversiblemappings(regionII) hastendedto obscurethis fact. To provide acomp!etesurveyof fig. 1.1, we alsoprovideexplicit examplesof area-preservingmappingsthat arenot reversible(region III) (cf. RobertsandCapel [1992a,b]).

We will be particularlyinterestedin reviewinguniversalityin reversiblemappingswith respectto twophenomenaassociatedwith the transitionto chaos.The first phenomenonconcernsthe breakingup ofKAM curvesarounda periodic orbit. Numericalstudieson many(reversible)area-preservingmappingshaveindicatedthat this processis characterisedby certainscalingnumberswhicharethe samefrom onearea-preservingmapping to the next (cf. Greene[1979],MacKay [1986]).It will be shown that thebreakupof invariant curvesaround a symmetricperiodic orbit of a general reversible mapping ischaracterisedby the samescalingnumbers.

The secondphenomenonstudiedis the period-doublingrouteto chaosin reversiblemappings.It isshownthat symmetricperiodic orbits period-doublewith the universalscalingnumbersassociatedwithperiod doubling in area-preservingmappings(cf. Benettinet a!. [1980a,b], Bountis [1981],Greeneet

1-3

~ A 13

—1~3Fig. 1.3. Conservativeand dissipativebehaviourcoexistingin areversibledynamicalsystem.Shownis thephaseportraitofa non area-preservingreversiblemapping L of theplanewith symmetry G: x’ y,y’ = x. The origin is an elliptic symmetricfixed point andis surroundedby invariantcurves.Thesecurves intersectthe symmetryline y= x andtheir symmetry with respectto reflectionaboutthis line is obvious.Also shownareelliptic symmetricsix-, seven-andeight-cyclesat the centresof theconcentric“rings” of six, sevenandeight islands.The KAM curvesandislandsaretypically associatedwith conservativesystems.Away from theorigin, the fixed point A at (I, —1) is attracting and its reflection GA = R at(—1, 1) is repelling. A trajectorythat spiralsin towardsA is shown; its reflectionby G spiralsoutwardsfrom R. An attractingfive-cycle is foundnearA anda trajectorythat spiralstowardsit is alsoshown.Thetrajectoriesof manyof thepointsnotenclosedby curvesaroundtheorigin or nearA escapeto infinity.

J.A.G. Robertsand G.R.W.Quispel,Chaos and time-reversalsymmetry 79

al. [1981]). This providesa quantitativesimilarity betweenthe period doubling in the two classesofmappings.MacKay [1982]haspreviouslyindicatedthat qua!itative!y the period-doublingbifurcationsofa periodic orbit in an area-preservingmappingand a syn~metricperiodic orbit in a reversiblemappingarethe same.The perioddoublingof (asymmetric)attractorsin reversiblemappingsis also shownto beuniversal,with the samescalings as found in dissipativesystems(cf. Cvitanovic [1989]).This perioddoubling can lead to the appearanceof a strangeattractor in reversible mappings. Because ofreversibility, the period-doublingcascadeof attractorsis accompaniedby a simultaneousperiod-doubling cascadeof repel!ersandthe appearanceof a strangerepeller.

Summarisingthe above,this reportwill showhowthe universalbehaviourdisplayedin conservativemappingscan be associatedwith universal behaviourof symmetric periodic orbits in reversiblemappings,and how the universal behaviourfound in dissipative mappingscan be associatedwithuniversalbehaviourof a reversiblemapping’sasymmetricperiodic orbits. Although many interestingquestionsconcerninggeneralreversiblemappingsof the planehaveyet to be answered,it seemstimelyto reviewandbring togetherwhat is knownto dateaboutreversiblemappings.Moreovermuchof thenew materialpresentedherecan be useful as a startingpoint for continuing their study.

The plan of this reportis as follows. In chapter2 we reviewin moretechnicaldetail thanabovethepropertiesof area-preservingmappings(and their generalisation— measure-preservingmappings) aswell as dissipativemappings.This is doneas part of a generalreviewof the propertiesof mappingsofthe plane.This discussionis includedso that the report is largely self-containedandcan be readwith aminimumbackgroundin nonlineardynamics.In chapter3 we expandon the conceptof reversibility fora mappingof the planeand discussthe consequencesfor the mapping’sdynamics.We showhow nonarea-preservingreversiblemappingsand non area-preservinginvolutions can be derivedfrom certainsecondorderdifferenceequations.In the last two sectionsof the chaptersomemethodsarepresentedfor decidingwhethera given mappingis not reversible.This leadsto explicit examplesof nonreversib!earea-preservingmappingsgiven in section3.4.

Large classesof non area-preservingreversiblemappingswith attractorsand repellers are con-structedin chapter 4. The methods provided are useful for creating reversible perturbationsofreversiblearea-preservingmappingslike theHénonmapping(1.23) so that (asymmetric)dissipativeandexpansivedynamical featuresare introduced without the loss of the conservative-!ikebehaviourassociatedwith symmetricfeatures.In chapter5 we concentrateon the conservativefeaturesof thesereversiblemappingsvia a studyof the breakingup of KAM curvesandof symmetricperiod doubling,as discussedabove.In chapter6 we concentrateon the dissipativefeatures,i.e. the attractors,in thesereversiblemappings.

In chapter7 we discussthe interplay of the conservativeand dissipative featuresof reversiblenonconservativemappingsvia a symmetry-breakingbifurcation (cf. also Post et a!. [1990a]).Thisbifurcation is a generalisationof a symmetry-breakingbifurcation studiedin reversibleconservativemappingsby Rimmer [1978,1983]. Finally, in chapter8 we summarisethe report and indicatesomepossibledirectionsfor future research.In appendixA, we give someintegrablereversiblemappingsofthe planewhich complementthe discussionin section5.1. In appendixB, we discussthe nonlinearstability analysisfor an elliptic fixed point which is used in section6.1.

2. Review of somepropertiesof mappingsof the plane

The interestin studyingmappingsof the planeis to observethe typesof possibledynamicsthat theycan have. In this chapterwe introducesomefundamentalconceptsassociatedwith the dynamicsof

80 l.A. G. Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry

mappings of the plane. Because we will be dealing in later chapterswith mappingswhich haveproperties similar to conservativesystemstogetherwith propertiessimilar to dissipativesystems,it isimportant to be familiar with the distinguishing dynamics of these two broad classes. At the sametimeit is importantto knowwhat propertiesa generalmappingof the planecan haveindependentlyof howthe areaof a region is changedwhenthe mappingis applied.This is becausethe reversiblemappingsofthis reportcannotbe classifiedas either area-preservingor dissipative.

Section2.1 reviewssometechnicalnotionsusedin the reportand is providedto makeit reasonablyself-contained.This sectioncan be passedover by thosereadersfamiliar with suchthings as invariantsets,classificationof periodicorbits by !inearisation,bifurcationof periodic orbits andPoincaréindex,and conjugacybetweenmappings.Generalreferencesfor discussionon someor all of thesetopicsinclude Arrowsmith andPlace[1990],Devaney[1986],GuckenheimerandHolmes [1983],Hsu [1987]andLichtenbergand Lieberman[1983].Section 2.2 is a discussionof measure-preservingmappings.

2.1. Definitions andproperties

A genera! mapping of the plane L: R2—+E~2 that assignsto a point x = (x, y) an image pointx’ = (x’, y’) can be written as

L:x’=f(x,y), y’=g(x,y), (2.1)

where f and g are arbitrary functionsthat may dependon one or more parameters.Most of themappings in this report will be diffeomorphisms (i.e., differentiable bijections with differentiableinverses).A diffeomorphism(2.1) is C” (moreprecisely,is of classC”) if all the partial derivativesoffandg up to andincluding orderk exist and arecontinuous.Many of the mappingsin this reportareinfact analytic,which implies that theyareC. The domainof L maybe thewholeplane~ x I1~,or if oneof x or y is an anglethe domainmaybethe cylinder l~x (— IT, IT] asin thestandardmapping(1.25). Inthis case,L is sometimesmorespecifically calleda mappingof the cylinder. If both x andy areangularvariables,L is a mappingof the torus: [— ir, ir] x [— IT, IT] with properidentifications.

The qualitative propertyof whethera planarmappingpreserves,contractsor expandsareais seenfrom the determinantof its Jacobianmatrix,

dL(x,y)z__(~~ ~ (2.2)

This determinant,DetdL(x, y), will be denotedby J = J(x, y). As is well known, the absolutevalueofJ gives the changein areaof a smallelementafter applicationof L. Mappingsof the planehavebeentraditionally classified by J. When J = ±1 throughout the plane the mappingis area-preservingorconservative[e.g. (1.23) and (1.25)]. Area-preservingmappingshavebeen studied well (see, forexample, MacKay [1982],Hénon [1983],MacKay and Meiss [1987],Arrowsmith and Place [1990,chapter6]). More generally,a conservativemappingis one for which we can write

J = m(x, y)Im(x’, y’), (2.3)

for somefunction m(x, y). In this casethe mappingL is called measure-preserving.Becausein theliteratureconservativemappingsareusuallytakento be area-preserving,wewill equate“conservative”


to area-preserving in this section.In section2.2 wewill discussmeasure-preservingmappingsandshowthat they possessmostof the propertiesof area-preservingmappings.

Most mappingsof the plane are not conservative.Nonconservativemappingsinclude dissipativemappingswhich have I JI <1 everywhere.On the otherhand,when J~> 1 everywhere,a mappingiscalled expansive.As an example,the two-parametermapping

x’=By, y’=—x+2Cy+2y2, (2.4)

or its equivalentforms, hasbeenstudiedwell. This mappinghasJ = B andso is dissipativewhenIBI <1(e.g., cf. Henon[1976],vander Weeleet a!. [1986],Cvitanovicet a!. [1988],Grassbergeret al. [1989])andexpansivefor IBI >1 (e.g., cf. Helleman[1980]).Most generally,mappingsof the plane haveaJacobiandeterminantthat varies throughoutthe planeand is locally contractingin someparts andlocally expandingin otherparts (cf. Chenciner[1983,1987], Gumowski andMira [1980]).

For diffeomorphisms,J is necessarilyof constantsign throughoutthe planebecausethe Jacobiandeterminantof a diffeomorphism is never zero (if it were, the inverse mapping would not bedifferentiable at a point; this follows from differentiating L ° L1 = Id, where Id is the identitymapping). When a mapping has J<0 or J>0 everywhere, it is called, respectively,orientation-reversingor orientation-preserving[Buck 1978, chapter8.4]. The propertyof orientationreversalhasthe following geometricconsequence:if we map somepointson a closedloop in the planewhich arechosenin clockwiseorderthentheir imagepointswill appearin counterclockwiseorderon the imageofthe loop. Orientation-preservingmappingspreservethe orderbetweenthe two setsof points.

Fromnow on we consideronly invertible mappings.The forwardorbit, or forward trajectory,of apoint x

0 = (x0, y0) under L is the set of points FL{xO} = {x1, 12, 13, . . .) such thatx,~1= Lx, withx~= (x,, y,). The backwardorbit of a pointx0 underL is the set of pointsBL{xo} = {x_1, x_2, X_3,. .

that,maps successivelyto x0, i.e., Lx_, = x_,~1.Thus FL{xo} = {L’x0, i E l~} and BL{xo} =

{L - ‘x0, i E1+}. Here L’ denotesL composedwith itself i times, i.e. L L °~ .. o L, and L - is itsinverse.The (full) orbit or trajectoryof point x~underL is the set of points { L ‘x0, i E7L}.

When we iterate an initial point 10 forwards or backwardsunder a genera!mapping (2.1), somecommonpossibilitiesare that either:

(i) we return to x~aftera finite numberof iterations,generatingalongthe way a finite set of distinctpoints (a zero-dimensionalset);

(ii) the orbit of x0 is aperiodicand appearsto fill a curve in the plane (a one-dimensionalset);(iii) the orbit of x0 is aperiodicbut appearsto denselyfil! a region of the plane— in this region two

initially close starting points typically have exponentiallydiverging orbits and the orbits are calledchaotic;

(iv) the orbit of x0 lies on a “strangeattractor” (seebelow);or the orbit of x0 is asymptoticto oneof thesepossibilities.Theorbit mayalso escapeto infinity if thedomainof the mappingis unboundedas whenit is the whole planeor the cylinder.

A set of points F is called invariant underthe mappingL if LF = F. An importantexampleof aninvariantset is a periodic trajectory,or n-cycle,which is a finite setof n pointsthatis mappedto itselfunderL, i.e., a set of pointsx, = {L’x0}, i = 1,. . . , n with L~X0= 10 and L’x0 ~ 10, 1 < i < n. Whenn = 1, x0 is calledafixed pointof L. Eachpoint of ann-cycle, calledaperiodicpoint of periodn, can beregardedas a fixed point of the mappingL “. That is, a periodic point of period n is a solution of theequations

82 JAG. Robertsand G.R.W.Quispel,Chaos andtime-reversalsymmetry

x~(x,y)—x=0, y~(x,y)—y=O, (2.5a,b)

wherethe functionsx,, and y,, are the x andy components of the mappingL~,formed by repeatedcompositionof L with itself. It is importantto realizethat anyresultderivedfor a fixed pointholdsfora periodic point after taking a suitablepowerof the mapping.A secondexampleof an invariant set is acurveC in the p!ane definedby the equation

I(x, y) = C, constant. (2.6)

If C satisfiesLC=C, i.e.,

I(x’,y’)=C (2.7)

whenever(x, y) satisfies(2.6), then it is called an invariant curve of the mapping.A third exampleof an invariant set in a mappingis an attractor. There is no universallyaccepted

definition of an attractor. An operationaldefinition is thatan attractoris a compactinvariantsetA towhich neighbouringpoints evolve under forward iteratesof L. As this working definition underliescomputerexperimentson attractorsin mappings,in the spirit of this report, it will be adoptedhere.Thedefinition can be mademoremathematicallyprecisein variousways (cf. Ruelle[1981,1989], EckmannandRuelle [1985]andMilnor [1985]) with differentemphasis;e.g.,whetheronerequiresall neighbour-ing points to convergeto A or almostall points — computational!yonecan typical!y only get afee! forthe secondpossibility. The setof all initial points10 in the planethat satisfyL’x0 —~A as i —+ ~ is calledthe basinof attractionB of A so thatL’B —+ A. We definearepellerR for a mappingL asa compactsetsatisfyingthe attractordefinition with L ~ L 1 andA R. That is, a repeller is an attractorfor theinversemappingL ~‘.

Becauseattractorsandrepellersinvolve, respectively,the contractionor expansionof an areain theplane, they cannot occur in area-preservingmappings. Attractors are the hallmark of dissipativemappings(e.g., Hénon [1976],Casdagli[1987,1988], Schmidt [1988])andcommonly occur in variousothernonconservativemappings(e.g.,Chenciner[1983,1987], Aronsonetal. [1983],Lauwerier[1986],Gumowski andMira [1980]).Attractorsandrepellersmaytakevariousforms; theseareattracting(orrepelling) periodic orbits, invariant curves and strangeattractors.A strange attractor (or strangerepeller) is an aperiodicattractor(repeller)which is characterisedby the fact that the motion on it ischaotic, i.e., sensitivelydependenton initial conditionswhich can be establishedvia the existenceof apositiveLyapunovexponent[EckmannandRue!le1985]. This chaoticmotion is often accompaniedbythe propertythat the attractorhasa fractalgeometricstructure.Althoughmostworkersusethe chaoticmotion criterion to define “strangeness”,some others place greateremphasison the self-similargeometryof the attractor[Grebogiet al. 1984; Holdenand Muhamad1986].

Any analysisof the dynamicsof the mappingof the plane (2.1) usually starts by looking for itsperiodic orbits andstudyingtheir stability. A fixed point10 of a mappingL is saidto be stableif foreverysmallneighbourhoodU of x~,thereexistsa neighbourhoodW of x0 suchthat all trajectoriesL’x,i >0, remainin U for x E W. Thereforethe ideais that onecan remainas closeas onelikesto the pointby choosinginitial pointsclose enoughto the fixed point. The fixed point is calledunstableif it is notstable.The linear stability of a fixed point is determinedby the eigenvaluesof theJacobianmatrix dLevaluatedat the point. We call thesethe eigenvaluesof the fixed point. Similarly, thelinear stability ofa periodic orbit of period n > 1 is found by evaluatingthe eigenvaluesof dL”, the Jacobianmatrix

l.A. G. Robertsand G.R.W. Quispel,Chaos and time-reversalsymmetry 83

of the mapping L”, at one point of the orbit. The matrix dL’ is called the return Jacobianof theorbit and is just the product of the mapping’s JacobiandL evaluatedat each point of the orbit[dL(x1). dL(x2) dL(x~)].

The eigenva!uesA of the matrix dLn, n � 1, aregiven by

A2 — Tr(dL~)A+ Det(dL~) 0, (2.8)

which hasthe solutions

A,2 = ~ {Tr(dL”) ± [Tr

2(dL”) — 4Det(dL”)] 1/2). (2.9)

The sum of the two eigenvaluesA1 and A2 is the traceTr(dL”); their product is Det(dL”), which is

called the returnJacobiandeterminantof the periodic orbit whenn> 1. SinceDet(dLfl) and Tr(dL ‘)arereal, the eigenvaluesarereal or complexconjugates.The classificationof ann-cycle is baseduponthe natureof the eigenvaluesof dL” andvarious casescan be considered(cf. Lauwerier [1986],Hsu[1987]).

The cycle is termedhyperbolic if neithereigenvaluelies on the unit circle in the complex plane.Inthis eventthe local behaviourof the mappingnearthe orbit is similar to the !inearisedbehaviour(theHartman—Grobmantheorem:the result holdsif L” is a C’ diffeomorphismin a neighbourhoodof theorbit, cf. Nitecki [1971]). In particular if IA,I <1 and IA2I <1 the periodic orbit is attracting. It isobviouslystableby the definitiongiven aboveandis morespecifically termedasymptoticallystable.Theconditionfor the n-cycle to be attractingwith IA1I <1 and IA2I <1 canbeexpressedin termsof Tr(dL”)and Det(dL”). We require

IDet(dL~)I<1, Tr(dL”)I <1 + Det(dL”). (2.lOa,b)

If a hyperboliccyclehasatleastoneeigenvalueoutsidethe unit circle, it is unstable.If IA1 I > 1 and

1A21 > 1 the orbit is repelling. Otherwise,A, and A2 satisfy IA,I > 1 and I A~<1 or vice versa,and arenecessarilyreal. A fixed point with such eigenva!uesis usuallycalled a saddlepoint. In the caseof ann-cyclewith sucheigenva!ues,eachpoint of the cycle is a saddlepoint of L”. It can beshownthat eachpoint of the n-cycle thenhastwo smoothinvariant curvespassingthroughit. Pointson onecurve,calledthe stable manifold W~,approachthe point of the cycle under iteration of the mappingL” whereaspoints on the other curve, called the unstablemanifold WU, approachthe point of the cycle underiterationof the inversemappingL “. The two curvesmayintersecttransversallyat a point, called atransversehomoclinicpoint (see Guckenheimerand Holmes [1983]).The orbit of sucha homoclinicpoint is calledahomoclinicorbit. Theexistenceof thesetransversehomoclinicpointsleadsto complexchaoticdynamicsin their vicinity. Similar complexity is associatedwith transverseheteroclinicpointswhich are the points of transverseintersectionof the stable and unstablemanifoldsof two differentsaddlecycles.

Whenboththe eigenvaluesof acycle lie on theunit circle in the complexplaneandarenot real (i.e.,A1 = e’°, A2 = e~0; ~ ~ 0, IT), the cycle is called elliptic becausein the linear approximationnearbytrajectories move on ellipsesencirclingthe pointsof theorbit. For a generalmappingof the planewithan elliptic cycle, the nearbymotion under the mapping L need not be the sameas in the linearapproximation.Instead,the nonlineareffects of L may causethe ellipses to degenerateinto inwardspiralsor outwardspiralsin whichcasethe cycle is an attractoror repeller.The situationis differentfor

84 l.A .G. Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry

orientation-preserving,area-preservingmappingswhere any periodic orbit with genuinely complexeigenvaluesis necessarilyelliptic since A,A2= Det(dL?~)= 1. In this case the KAM theorem forarea-preservingmappings [Moser 1962, 1973] applies. It states that under some differentiabilityconditionsand providedthe eigenvaluesarenot 3rdor 4th rootsof unity, thentypically eachpoint ofthe orbit is surroundedby infinitely manynestedclosedcurveslocatedarbitrarilycloseto the point andleft invariant by the mapping L~.On theseso-calledKAM curves, or KAM circles, the motion isquasiperiodic,i.e.,essentiallya rotationby anirrationalmultiple of 2 IT (for moredetails,cf. chapter5).Theeffect in two dimensionsof theseclosed invariantcurvesis to confinethe dynamicsof the mappingin the neighbourhoodof the cycle. This is becausethe trajectory of a point which is inside a closedinvariantcurvesurroundinga fixed point (eachpoint of the n-cycle is a fixed point of L”) mustremaininside the curve by continuity. Consequentlythe cycle is stable. KAM curvesarounde!liptic periodicorbits give area-preservingmappingstheir distinguishingphaseportraitof nestedislandswithin islands(suchas also appearin the centralpart of fig. 1.3). Betweenthesecurvesare thin bandsof boundedchaoticorbits associatedwith transversehomoc!inic points.

A third importantcasefor the eigenvaluesof a cycle occurswhentheysatisfyA1 = A2 = ±1. Thiscanhappen,for instance,in area-preservingmappingsin which casethe cycle is calledparabolic (cf. Siege!andMoser [1971,section31]). More generally,whenA1 = ±1or A2 = ±1,we havetransitioncasesforthe stability of a cycle. These casesfeatureprominently in an important aspectof the dynamicsofmappingsof the plane,whetherarea-preserving,dissipativeor otherwise,concernedwith the creationanddestructionof periodic orbits as amappingparameteris varied.Theeigenvaluesof a periodic orbitcan be usedto determinewhensuch eventscan occur.

A fixed point is isolatedfrom other fixed points or n-periodic points provided that neitherof itseigenvaluesis a root of unity. If this condition is satisfied at a particular value of the mappingparameter,it also guaranteesthat the fixed point persists (continuesto exist) in a small parameterinterval aroundthis value. These results follow from applying the implicit function theoremto eqs.(2.5) (cf. bossandJoseph[1980,appendixV.1]). A fixed point ‘0 = (x0,y0) is a solution of (2.5) foranyn � 1. The implicit function theoremguaranteesthat if x0 satisfies(2.5) at the parametervalue C0and the Jacobianmatrix of the left-handside of (2.5) hasnonzerodeterminantat (x0, y0, C0), thenthereis a uniquesolutionto (2.5) for someparameterinterval aroundC0andbox around(x0, y0). TheJacobiandeterminantcondition on (2.5) is equivalentto saying that at (x0, y0) and C0

Tr(dL~)~ 1 + Det(dL~). (2.11)

It is only when the sides of (2.11) are equal that the persistenceof the fixed point cannot beguaranteedand that the birth of an n-cycle from the fixed point may occur. The equality of theleft-handandright-handsides of (2.11) implies thatat leastoneof the eigenvaluesof the fixed point isan nth root of unity. As mentionedabove,importantspecialcasesarewhenn = 1 andn =2. When afixed point hasaneigenvalueequalto + 1 (the casen = 1), it cancollide with otherfixed pointsor givebirth to them. In a tangentbifurcation, fixed points‘are born from a part of the planewherepreviouslytherewereno fixed points. At birth thesefixed pointsmust haveat leastoneeigenvalueequalto + 1.Whena fixed point hasan eigenvalueequalto —1 (the casen = 2), a period-doublingbifurcationcanoccuranda two-cycle is created.Theperiod-doublingbifurcationsof fixed pointsin reversiblemappingswill be discussedin chapters5. and6. Note that in both of thesecases,as the bifurcation occursat atransition casefor the stability of the fixed point, the bifurcations are associatedwith a changeofstability.

l.A. G. Robertsand G.R.W. Quispel,Chaosand time-reversalsymmetry 85

A quantity thatplacessomerestrictionon thesort of bifurcationsthat canoccur in a mappingis thePoincaréindex [Hsu1980; Mira 1987, appendixA]. This is an integer, I, ascribedto an arbitrarilychosenclosedcurve in thep!aneanddefinedin termsof thenumberof revolutionsmadeby thevectorv= Lx — x as x movesoncearoundthe closedcurve.The sign givento this numberin order to obtainthe index is positiveor negativedependingon whetherthe rotationof the vectorvandthe traversalofthe curvearein the sameor oppositedirections.Theindex of a fixed point is definedas the index of aclosedcurveenclosingthisfixed pointand not containing,or intersecting,anyotherfixed points.For afixed point10 which is isolatedfrom otherfixed points[a sufficientconditionis (2.11)abovewith n = 1],the index can be relatedto its linearisation,i.e.,

TrdL(x0)<1+DetdL(x0)~I=+1, TrdL(x0)>1+DetdL(x0)~I=—1. (2.12a,b)

For example,anelliptic fixed point hasI = + 1, asdoesanattractingfixed pointwith 0< A1 <A2 < 1. Asaddlepoint with 0< A1 <1 <A2 hasindex —1. If a groupof fixed pointsis encircledby a curve, theindex of thecurve is given by the sum of the indicesof the fixed pointswithin. Significantly, providedthe c!osed curve doesnot intersectany fixed points, its index is constant.Therefore if a mappingparameteris varied and bifurcations of fixed points occur within a chosenclosed curve, the onlypossiblebifurcationsare thosefor which the indexof the curvebefore andafter the bifurcationsis thesame.This can also beapplied to bifurcationsof cycles,,as indicescan be assignedto them and relatedto their linearisationin a similar way (cf. Hsu [1980]).

Finally, we saysomethingaboutanimportantconceptthat arisesin the studyof mappings,namelythatof conjugacy[Nitecki 1971]. Two mappings(diffeomorphisms)L andM aresaidto be conjugateifthereexistsan invertible mappingP such that

M=P0L0P’. (2.13)

The conjugacy is classedas topological or differentiable accordingly as P is a homeomorphism(continuousbijection with a continuousinverse) or diffeomorphism. The processof rewriting theequationsof a mapping under a coordinatetransformationestablishesa conjugacy betweenthemappingin the new andold coordinatesystems.The Jacobianmatricesin the new andold coordinatesare relatedby

dM(x) = dP(LoP”(x)) dL(P~(x))dP1(x). (2.14)

The most important featuresof a mapping are those that are left unchangedunder conjugacy.Nitecki [1971]lists someof the invariantsof topologicalconjugacy.Forexample,if x~is a fixed point ofL thenPx

0 is a fixed point of M. For any~ the forwardandbackwardorbit of the point Px0 underMis the imageunder P of the orbit of ~0 underL, i.e.,

BM{Pxo} = P(BL{xO}), FM{Pxo) = P(FL{xo}). (2.15a,b)

More generallyif F is an invariant set for L thenP1” is aninvariant setfor M, andif I’ is an attractorthenso is P1”. Consequentlypossessionof an attractoris a conjugacyinvariant. Another importantconjugacyinvariant is the linear stability of a fixed point (or periodic orbit) becauseeigenvaluesof afixed point (or periodic orbit) are preservedunder differentiableconjugacy [eq. (2.14) becomesa

86 J.A.G. Robertsand G.R.W.Quispel,Chaos and time-reversalsymmetry

matrix conjugacy whenx is a fixed point of M]. It is not hard to show that orientationreversalandorientation preservation are alsomaintainedunderdifferentiableconjugacy.However,areapreserva-tion is in generalnot maintainedundersuch a conjugacyun!essthe Jacobiandeterminantof thecoordinatetransformationis constant,i.e., independentof x andy.

In the studyof mappingsit is often usefulto usecoordinatetransformations(conjugacies)to reduceamappingto a simpleor normalform. The ideais that theessentialnatureof the mappingis retainedbutthatthe studyof the normal form is easierbecausesomeof the complexityassociatedwith the termsofthe original mappingis removed.For example,from the theory of (Jordan)normal forms for a real2 x 2 matrixA, we know that thereexistsa realmatrix T suchthatA = TCT 1, whereCassumesoneofthe forms:

(A 0\ (A b\ (a —f3\~0 /L)’ ~0 A)’ ~~j3 a)~ (2.16a,b,c)

This result can be usedwhen expandinga mapping(2.1) in a Taylor seriesarounda fixed point.Assumingthat it is sufficiently differentiable, the mappingcan be written

x’ = Ax + By + F20x2 + F,,xy+ F

02y2+ F

30x3+ F

21x2y+ F,

2xy2 + F

03y3 +...,

(2.17)y’ = Cx + Dy + G

20x2 + G,,xy + G

02y2+ G

30x3 + G

21x2y+ G

12xy2+ G

03y3+...,

wherethe matrix (A B is the Jacobianmatrix evaluatedat the fixed point and this point hasbeen\C DI

shifted to the origin. The higher ordercoefficientsare relatedto the partial denvativesof f(x, y) andg(x,y) at the point. Becauseof the above, the expansioncan always be reducedvia a lineartransformationto a form

x’ = ax + by+f20x

2 +f1,xy +f02y

2+f30x

3 +f21x

2y+f12xy

2+f03y

3 +...,

(2.18)y’ = cx + dy+ g

20x2+ g,,xy+ g

02y2+ g

30x3 + g

2,x2y+ g,

2xy2+ g

03y3~

whereits linearpart (Jacobianmatrix) is oneof thethreeforms given. Now theeigenva!uesof the fixedpoint areprominent— theycan be readoff the diagonalin (2.16a,b) andarecomplexconjugatesa±if3in the caseof (2.16c). The norma! form (2.18), basedupon the linear part of the mapping,is thesimplestreductionwe can performon it. Some more powerful normal forms, basedupon the useofnonlineartransformationsto the original mapping,will be mentionedin chapter3.

2.2. Measurepreservation

In this sectionwe considermeasure-preservingmappings,a classof conservativemappingswhich aregeneralizationsof area-preservingmappings.A mappingL: x= (x, y)—+ x’ = (x’, y’) is calledmeasure-preservingif thereis a measure~ such that is(S)= /.L(L’(S)) for every measurableset S. Henceinvertible mappingsare measure-preservingif thereis a function m(x, y) suchthat

Jm(x,y)dxdy= f m(x’,y’)dx’dy’ (2.19)D L(D)

I.A.G. Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry 87

for any region of theplaneD. Thefunction m(x,y) is called a density.Fromthe formulafor changeofvariablein doubleintegrals,(2.19) is equivalentto (2.3),which canbe alternativelywritten usingvectornotation,

J= m(x)Im(Lx). (2.20)

Measure-preservingmappingswere studied, e.g., by Poincaré[1912]and Birkhoff [1913,1920,1927].Area-preservingmappingssatisfy(2.19) and(2.20)with m(x,y) 1. MappingsL = P’ o Mo Pthat areconjugateto area-preservingmappingsM are measure-preservingwith densitym(x,y) = DetdP(x).Conversely,ameasure-preservingmappingL with densitym(x, y) is conjugateto an area-preservingmappingif thereexists a coordinatetransformationP: X = u(x, y), Y = u(x, y) with Jacobiandeter-minant (au/ax)aulay — (aulay)aviax= m(x, y). A choice for the transformationP is [Moser 1965,Lemma2]: X = x, Y = f m(x, t) dt, which will be invertible in regionswherem(x, y) has constant sign,e.g., m(x, y) >0. Consequently,the class of measure-preservingmappingscan be thought of asessentiallythe classof area-preservingmappingsandmappingsconjugateto them, all with essentiallythe sameconservativedynamics.*)

Unlike area preservation,measurepreservationis maintainedunder (differentiable) conjugacy.However, whereasthe compositionof two area-preservingmappingsis still area-preserving,measurepreservationis not usually maintainedunder composition. Like the specialcaseof area-preservingmappings,measure-preservingmappingscannot possessattractors.This follows because(2.19) ingeneralimplies that no propersubsetV of the mappingdomain can be suchthat it containsits imageunderthe mappingi.e., V D L(V). The existenceof such a “trapping region” or shrinkingneighbour-hood is important in many definitions of an attractor (cf. Eckmann and Ruel!e [1985], Ruelle[1981,1989]).**)

Furthermore,a!! periodic orbits in anymeasure-preservingmappinghavereturnJacobiandetermin-ant ±1, which is a simpleconsequenceof (2.20). In particularif we look at the expansion(2.17)arounda fixed point of a measure-preservingmappingthenthis saysthat

AD—BC=±1. (2.21)

If the mapping is orientation-preserving,the right-handside of (2.21) is +1. From (2.16a,b, c) itfollows that in this casethe expansioncan be reducedto a norma!form (2.18) with a linear part givenby one of

(A 0 ‘~ (±1 b “~ (cosO —sinO’\ 222\0 A’)’ ~ 0 ±1)~ ~sinO cosO) ( . )

In the first casethe fixed point is hyperbolic (a saddle)andin the last case(0 � 0, IT) it is elliptic. Themiddle casecorrespondsto a transitioncasefor stability; typically b ~ 0.

Measurepreservationimplies infinitely manyrelationsbetweenthe coefficientsof the terms of anexpansionarounda fixed point. This can be deducedfrom Birkhoff’s study of measure-preserving

*) Notethat wewill sometimesalsocall measurepreservingamappingwith 5,, m(x,y) dx dy = ~ m(x’, y’) dx’ dy’ andI = —m(x)/m(Lx).~*) Note that a “measurepreserving”mappingin which thedensity is singularcould haveanattractorlocatedatthesingularity,e.g.,x’ = px,

y’ = py with 0<p < 1 hasdensity(xy)’ andan attractingfixed point at theorigin whosebasinof attractionis thewholeplane. Suchexamplesarenot really in thespirit of measurepreservationandperhapsshouldnotevenbeclassedassuch(mostof themeasure-preservingmappingsstudiedbyBirkhoff were suchthat their density waseverywherepositive).


Table 2.1Necessaryconditionsfor a mappingL to be locally measure-preservingarounda fixed point. Theconditionsrelatethe coefficientsof theexpansionof L, given in (2.18),when its linearpart hasone of thetypical canonicalforms

shown.

Linearpart Case Secondorder Thirdorder

(~A-’)’ A~‘±1 1 no condition (A — 1)(Ag,, + A1f,, —f,,g

11— 2f02g20) +f20f,,(A’ —2A’)

+ g,1g,2(2A’ — A’) = 0

(~~),b ~ 0 2 g,, +2120—2bg2, = 0 if g,0 = 0 then

3b2g,,— bg,, — 3bf,, +2f,,g

02+ 120111— 3bf20g,1= 0

(~“i), b oéO 3 no condition l2bg,,+ 4g,, + 121,,+ 2g~,+ 2f20g1,+22bf20g2,+ 10b2g~,

+ iof,1g,,+ llbg20g,1+4g20g02+ 12f~,= 0

mappingsin Birkhoff [1920].QuispelandCapel[1989]haveexplicitly calculatedsomeof theserelationsfor an orientation-preservingmeasure-preservingmapping. They take the expansionof the mappingaround the fixed point and write the JacobiandeterminantJ as a Taylor series with coefficientsdeterminedby the coefficientsof the mapping expansion.They expanda possible densitym(x, y)likewise(assumingdifferentiability of m to someorder) andsubstituteits seriestogetherwith that forJ(x, y) into the definition (2.20),or equivalently

J(x, y)F(x, y) = F(x’, y’), F(x, y) = [m(x, y)]~1, (2.23)

where x’ andy’ are input from the mappingexpansion.Equatingboth sides of (2.23) term by termleadsto “forma!” (i.e., without regardto convergence)necessaryconditions on the coefficientsof amappingin orderfor a density to exist to someorder. If a mapping satisfiestheseconditions to nthorderthenit is calledlocally measure-preservingto ordern. The conditionsup to third order arelistedin table 2.1 for typical linear parts of the mappingexpansion.Note that case1 of table2.1 yields theelliptic fixed-pointcaseon the right-handsideof (2.22) in its complexform if we takeA = e’°(0 ~ 0, ir).The real mapping expansionL: (x, y) -~ (x’, y’) with real linear part aroundan elliptic fixed point isconvertedto a complexform M: (X, Y)—+(X’, Y’) with complex linearpart via X= x + iy, Y= x — iy.

3. Reversibility andmappings of the plane

In this chapterweformally introducethepropertyof reversibility for a mappinganddiscusssomeofits consequences.As pointedout in theintroduction,reversibilityis a propertythat is quiteindependentof whethera mapping is measure-preservingor not and this gives reversiblemappingsthe scopetoincorporatea rangeof dynamicalbehaviours.Thegeneralfeaturesof reversiblemappingsarediscussedin section 3.1,and in section 3.2wegive a simple methodto createthem. In thelast two sectionsofthechapterwediscusstheproblemof howto recognisewhena given mappinghasthereversibility propertyandgive someexamp!esof mappingsthat are not reversible.


3.1. Somepropertiesofreversiblemappings

Recallingthedefinition given in section1.5 of the introduction,wehavethat a mappingof theplane~ ~2 given by (2.1) is reversible[Sevryuk1986] if thereexists a mappingG: R2 — ~2 satisfying

LOGOLG, (3.1)

where G is an involution, i.e.,

GOG=Id, (3.2)

andId is the identity mapping.G is calleda (reversing)symmetryof L. Equivalently,L can be written

as the composition of two involutionsL=HoG, HoH=GOG1d. (3.3a,b)

The reversibility property ensures that a mapping is invertible. In fact, from (3.1) and (3.2), theinverse mapping L 1 can be expressedas

L’_—GoLOG’=GOH, (3.4)

that is to say, the mapping and its inverse are conjugate,the conjugatingtransformationbeing thesymmetry G cf. (2.13) with M = L’, P = G and note G’ = G. This conjugacy and the associatedconjugacyinvanants[Nitecki1971] have important structuralconsequencesfor the dynamicsof themapping. In particular the reflection by G of the forward (backward)orbit of a point x

0 gives thebackward (forward) orbit of the point Gx0 [cf. eq. (2.15) with M = L

1 and note BL-,{GxO) =

FL{GxO} and FL-,{GxO} = BL{GxO)]. This hasbeenillustrated in fig. 1.2. Thus if the motion in onepart of theplaneis known, the motion in anotherpartof theplane(i.e., in the region reflectedby G)can be deduced.Note that the first equality in (3.4) expresses a conjugacy between L and its inverseindependentlyof whetherG is an involution. Mappingswhich satisfy (3.1) but do not requireG to bean involution as in (3.2) are called weakly reversible and have beenconsideredby Greene[1978],Arno!’d [1984],Arnol’d and Sevryuk [1986]and Sevryuk[1986],cf. also Lamb [1992].Although weconfine our discussionto reversible mappings,which have receivedmuch more attentionthan thesupersetof weakly reversiblemappings,someof thepropertiesbelow requireonly (3.1) insteadof (3.1)and (3.2).

As stressedin section1.5 the study of reversiblemappingshasbeenalmost exclusively directedtowardsmappingsthat arealso conservative.Nevertheless,thebasicreversibility property(3.1) p!acesno constrainton theJacobiandeterminantof L in genera!and doesnot imply measurepreservationofany sort [MacKay1982]. It is true howeverthat everydifferentiableinvolution is measure-preserving.This follows from differentiating(3.2) to give

dG(Gx) dG(x) = I, (3.5)

where I is the identity matrix, whence

90 J.A . G. Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry

DetdG(x) = IDet dG(x)Iu2IDetdG(Gx)I-1/2 sgn[Det dG(x)], (3.6)

with the’ densitygiven by m(x, y) = Detd G(x)ft’2 cf. (2.20). The productof two involutions, though,need not be measure-preservingbecause,as pointed out in section 2.2, measurepreservationis notnecessarilyretainedundercomposition.It is only recentlythatArnol’d andSevryukhaveprovedKAMtheorems for general reversiblediffeomorphismswhich can possessattractorsandrepe!!ersandso arenot measure-preserving[Sevryuk1986]. Explicit examplesof reversibleplanarmappingswith attractorsand repellerswill be given in thenext chapter.

Variouspropertiesof reversiblemappingswill now be listed. Someof thesewere consideredin thepioneeringworks of Birkhoff [Birkhoff1915; Birkhoff and Lifshitz 1945]. Further discussionof thepropertiescanbe foundin de Vogelaere[1958],Devaney[1976],Greene[1979],Greeneetal. [1981],MacKay [1982],Arno!’d [1984],Arno!’d and Sevryuk [1986]and Sevryuk[1986].Thesereferenceshowever, havenot consideredattractorsand repellersin reversiblemappings,with the exceptionofArno!’d and Sevryuk’s limited qualitative discussion. Although our interest here is in reversiblemappingsin R2 the majority of thepropertiescarryover to any dimension(e.g., cf. Kook andMeiss[1989]for a discussionof reversibility in 2n-dimensionalsymplecticmappings).

The first point that canbe madeabouta reversiblemappingis that it hasinfinitely manysymmetries.This follows becauseif G is a reversingsymmetryfor a mapping L, thenit is easilyverified that soarethe mappingsL’ o G for i an integer. The letter G can be used indiscriminately to represent,anysymmetry.The set {L’ o G) is called a family of symmetriesof L and togetherwith the set {L’} ofpowersof L it formsan infinite group[deVoge!aere1958; Piiia and JiménezLara 1987]. A mappingmaypossessmorethanone family of reversingsymmetries,in which caseit is calledmultiply reversible.For example,if a reversiblemappingL with symmetryG commuteswith a mappingM thenGo M andMo G satisfy (3.1) because

Lo(GoM)oL =(L0G0L)oM= G0M, Lo(M0G)oL = Mo(LOGOL)= MoG.(3.7a,b)

ThemappingsGo M or Mo G arereversingsymmetriesof L if they arealsoinvolutions (moregenerallyGo M and Mo G are all examplesof weakly reversingsymmetriesbecausethey satisfy (3.1) but notnecessarily (3.2) cf. Sevryuk [1986],Lamb [1992]).They are guaranteedto be involutions if Mcommuteswith G aswell aswith L and if M is itself aninvolution. Go M andMo G areindependentofthe symmetry family generatedby G provided that M ~ L’, i EZ. An important caseof doublyreversible mappingsis reversible odd maps, for which L(—x) = —L(x), i.e. L commuteswith themappingM 0 where

0:x’=—x, y’=—y. (3.8)

Reversibleodd maps havebeen discussedby Huiszoon [19831,MacKay [1984],Piiia and Cantoral[1989]and TanikawaandYamaguchi[1989].

The reversibility propertyis preservedunderconjugacybecause

(poLoP)o(PoGoP~)o(PoLoP~)= ~o GO’, (3.9)

which showsthat M = ~o L o P1 is reversible.Its symmetry~o G° P’ is conjugate to thesymmetryof

iA. G. Robertsand G.R.W.Quispel,Chaos and time-reversalsymmetry 91

L. Reversibilityis not in generalpreservedby composition.Howeverthecomposition(product)oftworeversible mappings that share a symmetry is reversible. In particular any power of a reversiblemapping is reversiblewith the samesymmetriesas themapping, i.e.,

L’oGoL’=G, iEZ. (3.10)

This readily follows from (3.1) and hasthe consequencethat

L’oGGoL’. (3.11)

Powersof a reversiblemappingprovide examplesof multiply reversiblemappings.This is becausetheone family of symmetriesfor L: { L’ o G}, i EZ, divides into the k distinct symmetry families forL’~:{L’~t~’oG} = {L’~’(L’oG)}, j = 0, 1,. . . , k — 1. For examp!e, if k= 2 and L is reversible withsymmetry G, then G and L o G are both symmetriesfor L2 but L o G is not in the symmetry family{L2’ o G} generated by G.

An importantpartof thestructurethat reversibilitygives to a mappingderivesfrom the setof fixedpointsof a symmetryG, denotedby Fix(G). From (3.5) it follows that the linear part of any symmetryevaluatedat oneof its fixed points is a linear involution. In thecommonlyencounteredcasewhenG isorientation-reversingthis meansthat dG hasthe form of thematrix

A=(’~ ~ p2+qr=1. (3.12)

An importantresult is that any involution is conjugate aroundoneof its fixed points to its linearpart(this holds in any dimension:“Bochner’s theorem”,cf. Montgomeryand Zippin [1955,section5.2J,Meyer [19811,Quispeland Capel[1989]).A simple way to seethis follows from the following identityfor any two involutions U andR: (U + R)oU = R o(U + R).Taking U = G, andR = dG evaluatedat afixed point x

0: Gx0 = ~ shows that G = [G + dG(x0)]’ odG(x0)o[G + dG(x0)] if G + dG(x0) isinvertible. The latter is alwaysinvertibleaboutx0 if G is C’, via the inversefunction theorembecauseits linearpart 2dG(x0)is nonsingu!ar.Becausethematrix A in (3.12)is similar to (~01), it follows thatarounda fixed point of an orientation-reversing involution G we can write

G=PoVoP’; V:x’=x, y’=—y. (3.13a,b)

Alternatively, at sucha fixed point G is also conjugateto anylinear involutionwhich is conjugateto V.Meyer [1981]showsthat if G in (3.13a) is area-preservingaswell as orientation-reversing,thenP canbe takenas area-preserving.

Finn [1974]hasshownthat the fixed point setof a C involutionof theplane,whetherorientation-preserving or orientation-reversing, is nonempty. *) For thecaseof an orientation-reversinginvolutionG, application of the reduction(3.13a) abouta fixed point, togetherwith the fact thatV in (3.13b)fixesthex-axis, thenimplies that G hasa curveof fixed points in theplanewhich doesnot intersectitself orterminate (it is sufficient for G to be C

1 to obtain this result; Finn assumesanalyticity of G andshowsthat the curve is also analytic). Such a curve of fixed points of G is called a symmetry line. Since

*) This is not necessarilytrue for involutions on othermanifolds.

92 JAG. Robertsand G.R.W.Quispel, Chaosand time-reversalsymmetry

virtually all of the involutionsof reversiblemappingsstudiedto date,aswell as thoseof themappingsof this report,areorientation-reversing,Fix(G) belowusuallyrepresentsa symmetryline. Ontheotherhand,if an involution is orientationpreserving,its fixed points neednot form lines,asthesimplelinearexamplex’ = —x, y’ = —y illustrates.BecauseL, reversible,hasinfinitely many symmetriesit also hasinfinitely manysymmetry lines. The symmetrylines of the family {L’ o G) canbe obtainedfrom thoseof G and H = L o G by taking their imagesunderpowersof L because

L’{Fix(G)) = Fix(L2’ o G), L’{Fix(L o G)) = Fix(L2”~’1o G), i El. (3.14a,b)

We now considerthe effect that reversibility has on the invariant setsof a mapping. We willcomplementsome of our discussionwith referencesto the phaseportrait fig. 1.3 of a non measure-preservingreversiblemapping, and to fig. 3.1 which shows some of its symmetry lines. (A colourversionof fig. 3.1 is found at the beginningof the report; a blackand white copy is providedhere.)

If F is an invariantset of a reversiblemappingL thenso is the set GE since

GF=(L0G0L)F=L(GF). (3.15)

For example,a point x0 is a fixed point of a reversiblemappingL = Ho G if and only if

Hx0=Gx0, (3.16)

and it is readily seenthat if; satisfies(3.16)thenso doesGx0. As a furtherexample,if the fixed pointx0 is a saddlepoint with stableandunstableinvariantmanifoldsW’(x0) andW”(x0), thenthe imagesoftheseinvariantsetsunderG are,respectively,theunstableandstableinvariantmanifolds of the fixedpoint Ox0, i.e., G{Ws(x0))= Wu(GXo) and G{W”(x0)} = Ws(Gx0). Incidentally, this relationshipmakesit easierto prove theexistenceof homoclinic andheteroclinicpoints in reversiblemappingsthan

/

~ Fig. 3.1. Fig. 1.3 with many of the trajectoriesremoved(acolour versionof this figure is foundat the beginningof the

report). The symmetry line y= x of G is drawn in black

~ X colour are shownthe first threeiteratesof the symmetryline~_>~ of G underL, which arethesymmetrylinesof L” G, L’ °Gand L’ G (cf. 3.i4a). Theseare colouredgreen, red andblue, respectively.All symmetrylines passthroughthesym-7 F3 together with the curved symmetry line of H=LoG. Inmetric fixed point at theoriginandtheothersymmetricfixedpoint at the otherintersectionof the lines of G andH. The

intersectionof the symmetry lines of L’ G and G accountsfor theexistenceofthesymmetricsix-cycle. l’his cyclehastwo

‘~ ‘~4 pointson eachof thesymmetrylinesof G anditsiterates.The______________________________________________________ seven-cycle,beingodd,hasapointon thesymmetryline of G

—13 andofH.

l.A. G. Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry 93

in general,e.g., if Ws(X0) intersectsFix(G), the point of intersectionis a homoclinic or heteroclinicpoint accordinglyasx0 = Gx0 or x0 ~ Ox0 [Devaney1984, 1988; Fontich 1990; HeagyandYuan1990].It also facilitates numerical calculation of the invariant manifolds, with application to transportproblems[Rom-Kedarand Wiggins 1990, sections6 and7].

Sincethe imageof aninvariantsetF underall thesymmetriesof the family { L’ ° G} is equalto GE,the invariant sets can be classified accordingto their invariancewith respectto the symmetry G. Aninvariant set F of the mapping which is also invariant under the application of a symmetry (i.e.,GE = F) is called symmetric. Symmetricsets, 1, can include invariant curves of the mappingandtrajectories(periodic or aperiodic), in which casewe talk of symmetric trajectories. An attractororrepellercannotbe symmetric(seebelow). A symmetrictrajectoryintersectsa symmetryline sinceforany xE1,

Gx=L’xEI~ forsomeiEZ (3.17)

implies that

G(L”2x) = L”2x or L’ o G(L ~‘2x)= L0’~2x, (3.18)

accordingly as i is even or odd [usingeq. (3.11)]. Conversely,any trajectory that intersectsthe

symmetry line of G (i.e., containsa pointy: Gy= y) is symmetricwith respectto G. This is becauseL’y = L’(Gy) = G(L~y), (3.19)

which shows that its forward orbit is the reflection by G of its backwardsorbit. Moreover since thepoints of the forward and backwardsorbits are iterationsofy E Fix(G) by somepositive or negativepowerof L, it follows from (3.14a) that everypoint of a symmetrictrajectorylies on a symmetryline.This is consistentwith the fact that the trajectory is symmetricwith respectto the entire family ofsymmetries.

The symmetricperiodic orbits are particularly importantexamplesof symmetrictrajectories.Everypoint of a symmetricperiodicorbit not only lies on a symmetryline, but it lies atthe intersectionof asubfamily of symmetry lines, namely the lines of L2’~G,i El if L~y= Gy = y [cf. eq. (3.14a)].Conversely,the point of intersectionof any two symmetry lines is necessarilya (symmetric)periodicpoint since

~ (L~~oG)o(L~oG)y=L~y=y, (3.20)

after using (3.11). Consequentlythis is how symmetricperiodicorbits arise(e.g., the six-cycle in fig.3.1) andcan be located(cf. Ichikawaet al. [19891,JungandRichter [19901,Richteret a!. [19901).Aspecialcaseof (3.20) occurswhenyE Fix(G) (i.e., p = 0) andyE Fix(L_2To G) (i.e., q = —2r), withL ry ~ y. This is equivalentto sayingthat y andL ry ~ y lie on thesamesymmetryline [cf.eq. (3.14a)],and (3.20)indicatesthaty is a periodicpoint ofperiod 2r.This implies that a symmetrictrajectorycanhaveat most two pointson a givensymmetryline, in which caseit is a cycleof evenperiod. It can beadditionally checkedthat for anysymmetricn-cycle,n even,with Gy= y thehalfway roundpoint L”2ylies on the same symmetry line [takex = y in (3.17) and note that Gy= y = Lay]. Therefore asymmetricperiodic orbit has evenperiod if and only if it containstwo points on a symmetry line.

94 iA. G. Robertsand G.R.W.Quispel,Chaosandtime-reversalsymmetry

Anotherspecialcaseof(3.20) occurswhenyand L”y are,respectively,fixed pointsof H = L ° G andG.In this casey is a fixed point of L2r+ 1 andnecessarilyhasodd period.Furthermoreany symmetriccyclewith odd period and with a point on the symmetry line of G hasa point on the symmetry line ofH= LOG (e.g., the seven-cyclein fig. 3.1).

The factthat everypoint of a symmetricperiodicorbit lieson a symmetryline andthat thesymmetrylines aresmoothcurvesis very evidentwhenonenumerically finds andplots manyperiodicorbits of areversiblemapping(see,e.g.,figuresin GumowskiandMira [1980]).Conversely,accordingto MacKay[1982,section1.1.4.5],Greenehasusedthe fact that periodicpoints of a perturbationof a reversiblemappingdid not appearto admit a smoothcurvethroughthem, to suggestthat theperturbationis notreversible. Tanikawa and Yamaguchi [1987,1989] have proved theoremsabout the ordering ofsymmetric periodicorbits of different periodson a symmetry line which arereminiscentof Sarkovskii’stheoremfor mappingsof the interval (cf. also Andrews [1989]).The simplestcaseof a symmetricperiodic orbit is a symmetricfixed point which is a point x0 that satisfies(3.16) andhasGx0 = x0. It isalso the generalcaseas everypointof a periodicorbit canbe discussedasa fixed pointof somepowerof L. From (3.14) it follows that everysymmetry line of a family of symmetry linespassesthroughasymmetricfixed point. Thus symmetricfixed points actasa focusfor the “web” that thesymmetrylinesform in the plane,cf. fig. 3.1 and Piiia and JiménezLara [1987],Piiia and Cantora! [1989].

Although theJacobiandeterminantJof an arbitrary reversiblemappingvariesthroughouttheplane,symmetric periodic orbits are significant in that their return Jacobiandeterminantequals±1. Thisfollows from differentiating(3.10) with i = n and usingL~y= Gy= y,

dL~(GL~y)dG(L~y)dL~(y)= dG(y) ~ {Det[dL~(y)]}2 = 1. (3.21)

When the componentinvolutions H and G of L = Ho G are orientation-reversing,the Jacobiandeterminantof L is necessarilypositive (L is orientation-preserving)andthe returnJacobiandetermin-antof symmetricperiodicorbits equals+ 1. This suggestssomeapproximatearea-preservationpropertyafter traversalof the periodic orbit, or equivalently a local “almost-area-preserving”environmentaroundeachpoint of an n-cycle when consideredas a fixed point of L”. Work done by Quispel andCapel [1989],which will be discussedin more depth in section3.3, shows that this “almost-area-preserving”property associated with symmetricperiodicorbits oftenextendsto higherorders.

The linear stability analysisof symmetricperiodic orbits in orientation-preservingreversiblemap-pings (on which we concentratein this report) is thesameasfor cyclesin measure-orarea-preservingmappingsso that cycles are generically elliptic or hyperbolic with real reciprocal eigenva!ues.Thenorma!formsfor thelinear partof thereturnmappingL” at a point ofthen-cyclearethesameasthosefor a measure-preservingmappingarounda fixed point, e.g.,

(A 0 ‘~ (±1 b ‘~ (cos 0 —sin 0 3 22\0 A’)’ \ 0 ±1)’ \sin0 cosO

The KAM theoremsof Arnol’d and Sevryukshow that the nonlinearstability of symmetricellipticperiodic orbits is also the sameasthat in area-preservingmappings,i.e., a symmetricelliptic periodicorbit is in generalstablebecauseit is surroundedarbitrarily closelyby nestedclosedinvariantcurvesthat are symmetric. This “conservative”behaviourof reversiblemappingsassociatedwith symmetricsets will be discussed in chapter 5 and is very evident from the curvesand islandsshownin fig. 1.3.

When an invariantsetF of a reversiblemappingwith symmetry0 is not symmetric, therearetworemaining possibilities: (i) F fl OF = 0, the empty set; or (ii) F fl GF = V, V nonempty.In the first

I.A.G. Robertsand G.R.W. Quispel,Chaosand time-reversalsymmetry 95

case, we call the invariant set F asymmetricanddenoteit by [~. In this case GFa is an invariant setdistinct from F~.In thesecondcase,V is a strict subset of F and GE, and is a symmetricinvariantset.Hencein this casethe union of F and GE, which is also an invariant set, is composedof a symmetncinvariant subsetand a pair of asymmetricinvariant subsets.In general then an invariant set of areversible mappingis symmetric,asymmetric,or composedof symmetricinvariant subsetsandpairsofasymmetricinvariant subsets.Periodic orbits are necessarilysymmetric or asymmetric.Attractors,whetherthey be periodic orbits or not, are necessarilyasymmetricso that an exampleof a 17, is anattractorA with basin of attractionB. Then it is not hard to showthat GA is a repellerwith basinofrepulsion GB. This follows from the fact that L and L ~1 are conjugate[cf. eq. (3.4)] and GA ~sthereforean attractorfor L 1 becausepossessionof an attractoris invariant underconjugacy.

Therefore,attractorsand repellerscomein pairs in reversiblemappings,asshown by the attractingfixed point in fig. 1.3 andits reflection,which is a repellingfixed point. In the simple caseof a periodicattractorA, this can also be understoodby noting that theperiodicorbit GA alwayshasthe oppositelinear stability to A, i.e., the eigenvalues of its return Jacobianarethe reciprocalsof thoseof A. Thispropertywasstatedin Politi et al. [1985,1986a,b] and independentlyprovedin Feingoldet a!. [1988].At the level of fixed points it saysthat if x0 is an asymmetricfixed point [i.e., it satisfies(3.16) withOx0~ x0] and haseigenvaluesA and p., then the fixed point Ox0 haseigenvaluesA

1 and p.~. Thisfollows because(3.4) is a differentiableconjugacy if G is a diffeomorphism,and thus the matricesdL - ‘(Gx

0) and dL(x0) share the same eigenvalues. But dL - ‘(Gx0) is the inverse of the matrixdL(Gx0) andhencetheeigenvaluesof dL(Gx0) arereciprocalsof thoseof dL(x0), i.e.,A’ mustbe aneigenvalueof dL(Gx0) if A is an eigenvalueof dL(x0).*) As usual the result for periodic orbits(n-cycles)follows by using theconjugacybetweenL~and L “. Note howeverthat asymmetricperiodicorbits mayhave return Jacobian determinants equal to 1 and be linearly (elliptically) stable, and still beattracting or repelling — see chapter 4.

An attractor or repeller in a reversible mapping cannot intersect a symmetry line because this wouldimply that they are symmetric.**) Then A = GA, andA or GA would needto be an attractorand arepeller at the same time which is impossible. This explains why symmetric sets cannot be attractors orrepellers. Symmetry lines in the vicinity of an attractor or a repeller line up parallel to one another incontrast to their behaviour near symmetric periodic orbits cf. fig. 3.1, and chapter 6 where thedissipative behaviour in reversible mappings associated with their asymmetric sets will be discussed.

3.2. Generatingreversiblemappingsand involutionsfrom symmetricsecondorder differenceequations

The previous section was largely theoretical in nature and gave an indication of the many propertiesthatarisewhena mappingL satisfies the reversibility condition (3.1). Of immediate practical interest ishow difficult it is to find mappings that satisfy this property. Somereversible mappings arise naturally

*) This relation holds in any dimensionfor reversiblemappings. Of courseit also applies to symmetricfixed points x, = Gx, or symmetric

periodic orbits and showsthat their linear stability is similar to symplecticmappingsin that eigenvaluesmust come in reciprocalpairs. Formeasure-preservingmappings,the restriction that the eigenvalueshave a product of 1 is less severe.In higher dimensions,periodic orbits inmeasure-preservngsystemsaretypically unstable.The importanceof reversibility in thesesystemsis that it can stabilisesymmetricorbits, givingthesesystemstheir only stableorbits. Non measure-preservingsystems,on thecontrary,typically haveattractors.Theimportanceof reversibilityinthesesystemsis thatit can give riseto theconservativebehaviourassociatedwith symmetricperiodic orbitswhilst still allowing coexistingdissipativeand expansivebehaviour.

~ Note that for anaperiodicattractortheproof (3.19) needsto be modified to accountfor the fact that theattractoris not thetrajectoryofsomepoint. Theproofcan bedoneby assumingthatthereis a denseorbit on theattractor,i.e., anorbit of somepointof theattractorwhichcomesarbitrarily close to eachotherpoint of it.

96 J.A.G. Robertsand G.R.W.Quispel,Chaosand tune-reversalsymmetry

from a physicalproblem(e.g.,from a surfaceof sectionof a reversibledifferential equation).Thepointof this section is to showthat manyreversiblemappingsand their componentinvolutionscan befoundby consideringsymmetricsecondorder difference equations,using the fact that any secondorderdifferenceequationcan be rewritten as two coupledfirst order differenceequations.The reversiblemappingsfoundin this wayincludethe importantformsof area-preservingmappingpreviouslystudiedin the literatureand some non measure-preserving mappings that will be discussedin later chapters.

A lot of the mappingsof the plane that have been studied arise from secondorder differenceequations,

(3.23)

which have the propertythat they canbe explicitly solvedfor x,, + 1 as a function of x,, -1 andx,,, i.e.,

x~÷1= F(x~,1,x~). (3.24)

By using the identification

(x~1,x~)~-s.(x, y)=x, (x~,x~÷1).~-s.(x’, y’)=x’ , (3.25a,b)

we can associate (3.24) with the mapping of the plane

L:x’=y, y’=F(x,y). (3.26)

(Note that some authors make the alternative identification (x~ 1’ x~)~ (y, x); for higher orderrecurrencerelations and their relation to mappingscf. Angenent [1990]).Conversely, many, butcertainly not a!l, mappingsof the planex’ = f(x, y), y’ = g(x, y) can be written as a secondorderdifference equation. Certainly a sufficient condition for this to be possible is for x’ = f(x, y) to besolvableexplicitly for y(x,x’) or for y’ = g(x, y) to besolvableexplicitly for x(y, y’), e.g.,if f(x, y) islinear in y or g(x, y) is linear in x. A secondorder difference equationcan then be createdonsubstitution for the solved variable into the other equation of the mapping (with an appropriateidentification).

Onewayto createreversible mappingsof theplaneis to considersecondorderdifferenceequationsthat are symmetric, i.e.,

H(x~~1,x,,_,,x~)= H(x~1,x,,~.1,x~), (3.27)

which ensuresthat (3.23) can be solvedfor x,,_1 with the same dependence on x,,,~1 and x, as x,,~1 hason x,, andx, (assumingstill that we can explicitly solve for x,, +~ i.e.,

~ (3.28)

In this case the differenceequation(3.23) is invertible (backwarditeratesareuniquely specified)andwe can show that the mapping (3.26) derived from such a symmetricdifferenceequationis reversible.Taking

S:x’=y, y’=x, (3.29)

l.A. G. Robertsand G.R.W.Quispel,Chaos and time-reversalsymmetry 97

we find that

L0S0L(x, y)=(y, F(F(x, y), y)). (3.30)

From (3.28) with (3.25a,b) we have

F(F(x, y), y) = x, (3.31)

and hence

L0S0L=S. (3.32)

SinceS is an involution, themappingL is reversible[ci. eq. (3.32) with (3.1)1. It follows that LoS isalso an involution

LOS:x’=y, y’=F(y,x). (3.33)

Thisshowsthat if amappingof the planecan be reducedto a secondorder differenceequation that issymmetric,then it is reversible.

If, as well as being symmetric, the difference equation(3.23) is such that

H(—x~~1,—x,,1, —x~)=±H(x~~1,x~_1,x~), . (3.34)

thenF(—x~1,—x~)= —F(x~,,x~),and thederivedmappingL in (3.26) is oddsinceit commuteswith

0: x’ = —x, y’ = —y. (3.35)

Thenthe mapping T=SoO=OoS, i.e.,

T:xl=_y, y’=—x, (3.36)

is a symmetryof L distinct from the family generatedby S [cf. (3.7a,b) with 0 = S, M = 0 and note

that T is an involution]. As a resultL ° T is also an involution, whereL0T:x’=—x, y’=—F(y,x). (3.37)

Wehave used symmetric, and symmetric-odd, second order difference equations to create variousinvolutions L o S andL o T. We nowpresent some examples of difference equations (i~E)togetherwiththeir associatedreversiblemappingsof the planeL and involutionsL o S andL o T (the conditions onthe functionslisted after L o T guaranteethat L is odd). All the involutions obtainedareorientation-reversing.

Example1

i~E: ~

Mapping: L:x’=y, y’=—x+f(y); (3.38)

98 J.A.G. Robertsand G.R.W. Quispel,Chaosand time-reversalsymmetry

Involution: LOS: f =x, y’ = —y+f(x); (3.39)

Involution: L ° T: x’ = —x, y’ = y —f(x), f odd. (3.40)

Example 2

iXE: ~ +x~,.1)f2(x~)+f,(x~)=0.

Mapping: L:x’=y, ~ ; (3.41)

Involution: LOS:x’=x, ~ ; (3.42)

f (x) — yf (x)Involution: L ° T: x’ = — x, = 1 2

—f2(x)+ yf3(x)

f1/f2 odd, f31f2 odd. (3.43)

Example 3

i~E: ~

~ m2_3fg<0.*)

Mapping:

I x’ = y,

L:~(yl3+ x3)f(x’) + (y’2 + x2)m(x’) + (y’ + x)g(x’) + h(x’) = 0; (3.44)

Involution:

LoS:{(y?3+’y3)f(x)+(yP2+y2)m(x)+(yl+y)g(x)+h(x)0; (3.45)

Involution:

I x’ = —x,

Lo T:~(yP3— y3)f(x’) + (y’2 + y2)m(x’) + (y’ — y)g(x’) + h(x’) = 0;f/h, g/h odd, rn/h even. (3.46)

The mappingL in example1 is an area-preservingmappingand is in fact oneof thestandardformsfor area-preservingreversiblemappings.We will cal! themappingform (3.38) the McMillan form (afterMcMillan [1971]).Equation(1.23) is the McMil!an form of theHénonmapping. Its phaseportrait isshown in fig. 3.2.

* Thepointof this condition is thatit guaranteesthatthereis auniquesolutionx~,* to H(x,, ~,x~1,x,,)= 0 by ensuringthat H(x~ x~)

is a monotoniccubic polynomial in x~,1.


_______________________1•0_______________________

Fig. 3.2. Phaseportrait of the Hénon mapping in McMillan form: x’ = y, y’ = —x + 2Cy+ 2y2, when C= 0.3. The symmetry lines y= x and

y = Cx+ x2 areshown.Theorigin is anelliptic symmetricfixed point. Another fixed point whichis alsosymmetric,but hyperbolic,is at theotherpoint of intersectionof thesymmetrylines, i.e., at (1 — C, 1 — C) = (0.7,0.7).

Table 3.1 presents,as well as the McMillan form, two otherwell knownandwell studiedformsofarea-preservingreversiblemappings.The generalisedstandardform is the compositionof one involu-tion of the type (3.39)andoneinvolution of the type (3.40).Theseinvolutionsareorientation-reversingand area-preserving.The Chirikov—Taylor or standardmapping(1.25) is of generalised standard formwith f(y) = y andg(x’) = (K/2ir) sin2irx’. Becausef(y) andg(x’) areboth odd, it canbe decomposed

Table 3.1Threeclassesof area-preservingreversiblemappingsL = HoG.

Involutions

H GClass MappingL [symmetryline] [symmetryline! Fixed points

Generalisedstandard x’ = f(y) + x x’ = —x x’ = —x — fly) (x0, yo): g(x0)= f(y,) = 0

y’=g(x’)+y, godd y’=y—g(x), godd y’y[x=0l [x=—f(y)12]

Generalisedstandard x’ = fly) + x, f odd x’ = x x’ = x + f(y), f odd (x0, y0): g(x0)= f(y0) = 0y’g(x’)+y y’=—y+g(x) y’=—y

[y=g(x)12l [y°°O]

McMillan x’=y x’=x x’=y (x,,x0):f(x0)—2x,0y=—x+f(y) y’=—y+f(x) y’=x

[yo~f(x)/2] [yxl

DeVogelaere x’=f(x)—y x’=y+f(x) x’=x (x0,0):f(x0)—x00y’=x—f(x’) y’x—f(x’)

[y=x—f(x)] [y=°l


into the productof involutions in two ways [thisdoublereversibility occursbecause(1.25) commuteswith (3.35)— cf. (3 .7a, b)]. The generalised standard form is important because it includes the other twoforms in table 3.1 via coordinate transformations. A mapping in McMi!!àn form (3.38) can be written ingeneralised standard form by using the transformationx —~x, y—~x — y. Figure 3.3 shows a phaseportrait of Hénon’smapping in generalisedstandardform. Mappingsin de Vogelaereform canalsobetransformedto generalisedstandardform becausetheycanbe transformedto McMillan form via x —~y,

y —~x — f( y). Although it is not possibleto convertan arbitrarymappingin generalisedstandardforminto McMillan form, manymappingsof physicalinterestcanalwaysbewritten in McMillan form sothatit is practicallya very important form (cf. Laslett [1978,1986]).

The involutions of examp!es2 and 3 above are not in general area-preserving.Their Jacobiandeterminantsaregiven by ±ay’ / äy, which canbefoundby implicit differentiationin example3. Sinceareversiblemapping is simply a combinationof involutions, any non area-preservinginvolution canbecoupled to another or to an area-preservinginvolution to form a non area-preservingreversiblemapping.This approachwill be usedin the next chapterwheremappingsthat arenot area-preservingand not measure-preserving,like that depictedin fig. 1.3, will be constructed.

3.3. Testingfor reversibility and local reversibility

In theprevioussectionswediscussedreversibilityand its consequencesandthengavesomeexamplesof mappingsthat arereversible. In this and the next sectionwe continueto explorethepropertiesofreversibility but with a view to answeringthe question: how does one tell if a given mapping isreversible?Since reversibility givesamappinga lot of dynamicalstructure,it is desirableto haveatestfor it andto explicitly find a reversingsymmetryfor the mapping.

Certainlysomeof the propertiesof reversibility canbe usedasnecessaryconditionsto showthat a

_______________________1~0_______________________Y

Fig. 3.3. Phase portrait of the Hénon mapping in generalisedstandardform: x’ = —y + x, y’ = 2x’(l — C — x) + y, whenC = 0.5. Thesymmetrylines are thex axis and the parabolay = x (1 — C— x). The origin is an elliptic symmetricfixed point and (1 — C,0)= (0.5,0) is a hyperbolicsymmetricfixed point.

J.A.G. Robertsand G.R.W.Quispel,Chaos andtime-reversalsymmetry 101

given mappingis not reversible.Oneof the mostapplicableis the reciprocal-eigenva!uespropertyofpairsof asymmetricfixed points(or periodicorbits) in reversiblemappingsthatwasmentionedaboveinsection3.1. Firstly, considera given non measure-preservingmapping;typically it hasa fixed point atwhich J ~ ±1. If this mappingwould be reversible,sucha fixed point is necessarilyasymmetricandsowould havea partneringfixed point with reciprocaleigenvalues.If, in the given mapping,thereis nosuch partner (which would be the typical case for an arbitrary mapping) then the non measure-preserving mapping can be declarednot reversible. Considernow a measure-or area-preservingmappingwhich is also orientation-reversing.Now every fixed point hasJ= —1 andevery odd-periodicorbit hasa returnJacobiandeterminantof —1. A possible symmetricfixed point in sucha mappingmusthaveeigenvalues{ 1, —1 } to be consistentwith A and A 1 bothbeingeigenvalues.Consequently,most fixed points in orientation-reversingconservativemappingscan be ruled out as beingpossiblesymmetric fixed points, and then the mapping can be excluded as being reversible when possibleasymmetricpartnerswith reciprocaleigenvaluescannotbefoundfor suchpoints,cf. RobertsandCapel[1992b].On the other hand, for an orientation-preservingmeasure-or area-preservingmapping,applying a reversibility testat the level of fixed pointsproves a lot less effective becauseevery fixedpoint hasJ = +1. If a partnerwith reciprocaleigenva!uesdoesnot exist for a fixed point or periodicorbit, themost that canbe concludedis that the fixed point orperiodicorbit is necessarilysymmetricifthe mappingis reversible[in two dimensionsall possiblelinearisationswhenJ= + 1 at a fixed point areconsistentwith it being a symmetricfixed point in a reversiblemapping,cf. (2.22) and (3.22)]. Notethat whenJ= + 1, an asymmetricpartnerof a fixed point hasthesameeigenvaluesasthe fixed pointbecausethe eigenvaluesof any fixed pointsare themselvesreciprocalsof one another.

An explicit exampleof an orientation-preservingarea-preservingmappingthat may notbe reversibleis

R:x’=x+y+l—cosy, y’=y—Csinx’— C(1—cosx’). (3.47)

This mappingwas createdby Rannou[1974].Its form is similar to the generalisedstandardform oftable3.1 but hereneitherf(y)= y + 1— cosy norg(x’)= —Csinx’ — C(1 — cosx’) is odd, so that it isnot known whetherit is reversible.An exceptionoccurswhen C = 0 in which caseg(x’) 0 is oddtrivially andthe mappingis reversible.Themapping(3.47)is a mappingof the torus, — ir < x � ir and— ~r< y � ir, sothat x andy aretakenmod2ir. It hastwo fixed points at (0, 0) and(— irI2, 0) which areelliptic for 0< C<4 and —4<C <0, respectively.Becausetheseparameterintervalsaredifferent, thetwo points are not an asymmetricpair if the mapping is reversibleand so can only be symmetric.Aphaseportrait of Rannou’smapping (cf. Rannou[1974])shows no obviousglobal symmetrylike thatseenin the reversiblemappingsin figs. 3.1,3.2 and3.3,but this apparentlack ofsymmetryis also foundin manyreversiblemappingsthat do nothavea linear reversingsymmetry(i.e., it is hard to “see” theeffect of nonlinearreversingsymmetries).

Although not studying reversibility as such,Rannoucreated(3.47) to avoid “hidden symmetries”obtainedin numerical resultson other mappingswhich were reversible.*) No such symmetry in theresults was observedfor (3.47). This appearsto be the reasonthat Rannou’smapping has beensuggested in the literature as not being reversible (cf. Greene et al. [1981],MacKay [1983a1;alsoGrebogiandKaufman[1981]).In this andthenext sectionwe describea methodfor telling whether

*) Rannou’sinterestwasto study theeffect of round-offerrorsin mappingsof theplane.This wasdoneby studying(3.47)on alattice so thatx

and y took a finite numberof valuesand themappingcould be definedon the integers(cf. Rannou[1974]and also Hénon [19831,section9]).


nonconservativemappings,as well as a specialclass of area-preservingmappings,are not reversible.The reversibility or nonreversibilityof Rannou’smapping,however, remainsan open question.

A mapping is called locally reversibleby MacKay if it is reversiblein someregionof theplane(cf.MacKay [1982,section1.1.4.5]).That is, thedefining relation L ° G o L = 0 holds for someinvolutionG in someregionof theplane,or, equivalently,themappingcanbe writtenasthe compositionof twoinvolutions in someregion.Obviously all reversiblemappingsasdefinedin section3.1 (which might becalled “globally” reversiblemappings)are locally reversibleeverywhere.The converseneednot betrue.

The concept of local reversibility we use here is that of local reversibility to order n abouta fixedpoint. It involvesfinding outwhethera mappingis reversiblein a Taylorexpansionto ordern aboutthefixed point. This ideawasintroducedin Quispeland Cape! [1989]andwe follow the treatmentgiventhere.The first step of this methodis to identify a fixed pointp of a given mappingL as necessarilysymmetric if the mapping would be reversible by comparing its eigenvalues to those of other fixedpoints, as outlined above. This point is therefore a commonfixed point of the mapping and any possiblesymmetry.

Assumethen that the mappingL hasa Taylor seriesexpansionof ordern in a neighbourhoodofp — asufficient condition is that L is C”, k> n. This obviouslyincludesmappingsthatareanalytic in aneighbourhoodof thepoint. ThenconsidermappingsG which alsohaveTaylorexpansionsof ordernaroundthe fixed pointp andareinvolutory to ordern in a neighbourhood of p, i.e., G ° G =Id + 0,,+ 1’wherethe0,,+ notationstandsfor termsof ordern + 1 andhigher.The classof mappingsG obviouslyincludesal!C” involutionsoftheplane,k> n, andin particularanalyticinvolutions.SubstitutingsuchaG into L o Go L = G andequatingboth sidesof this relationto ordern leadsto necessaryconditionsonthe coefficientsof the expansionof L up to ordern in order for a G to exist. If theseconditionsaresatisfied, andthereforeto ordern thereexists a G, the mappingis called locally reversibleto ordernaroundthe fixed point. On the otherhand, if theseconditionsarenot satisfiedthenlocal reversibilitywith respectto the classof mappingsG is ruled out. Consequently,global reversibility of L with respectto all C~cinvolutions,k> n (aswell as all analyticinvolutions) is alsoruledout. It is in this sensethat wethen call L not reversible.

Note that the necessaryconditions for local reversibility derived in the way just describedarepreciselythe sameas thosefor reversibility of a “formal” diffeomorphismwith respectto a “formal”reversingsymmetry.Here “formal” refers to a mapping or involution defined by an infinite powerseriesexpansionabouta fixed point, without regardto convergenceof theseries.Analyticmappingsorinvolutionsare a specialcaseof formal oneswhenthe seriesdoesindeedconvergein a neighbourhoodof the point (for somemorediscussionof “formal” propertiesof diffeomorphismssee,e.g.,Broer andTakens[1989]). It is readily verified that local reversibility as describedhasmany of the importantpropertiesof global reversibilitydiscussedin section3.1. Forexample,if L is locally reversibleto ordern with G beingthe reversinginvo!ution to order n, then we have:

(i) {L’oG} are also reversinginvolutionsto order n;(ii) L’ is locally reversible to order n, with reversinginvolution G;(iii) local reversibility is maintainedunder coordinatetransformations,i.e., ~o L ° P1 is locally

reversibleto ordern with ~o G P1 being the reversinginvolution to order n.Furthermore,the various alternativeways of expressingthe global reversibility property are also

equivalentat the level of local reversibility to order n. For example,sinceareversiblemappingcan bewritten

LU0R, (3.48)


where U andR areinvolutions, andsincelocally abouta symmetricfixed point an orientation-reversing

involution U can be written asU=P0V0P’, (3.49)

with V givenby (3.13b),the following relationholds for a reversiblemappingL abouta symmetricfixed

point:

P~oL=VoP1oR, (3.50)

for somecoordinatetransformationP’ and involution R. In the commoncasewhenL is orientation-preserving, R is necessarily orientation-reversing if U is orientation-reversing.

The formulation(3.50)can be usedto investigatelocal reversibility conditions (cf. Brown [1991]fora different approach).Assumethe (possiblysymmetric)fixed point to beat theorigin andexpandL, Rand P1 “formally” around this point in Taylor series.The expansionfor the mapping L, assumedorientation-preserving,can be chosento be

x’ = ax + by+f20x

2 +f11xy +f02y

2 +f30x

3 +f21x

2y+f12xy

2+f03y

3 +...,

(3.51)y’ = cx + dy + g

20x2 + g

11xy+ g02y

2 + g30x

3 + g21x

2y + g12xy

2 + g

03y

3 ~

with ad — bc = 1 and the linearpart,which wewill write as (a, b, c, d), given by oneof eqs.(3.22).Thepossible involution R is assumedto be orientation-reversing,in line with the common casethatreversibility of mappings is with respect to orientation-reversinginvolutions. Consequently,theexpansionfor R must havea linear part of the form (3.12).

Following the generalproceduredescribedabove,the expansionsfor L, R andP’ aresubstitutedinto (3.50) and equatinglike terms on eachside leadsto relationsconnectingtheir coefficients.Thecoefficientsfor R and P1 can be eliminated to give necessaryconditions on the coefficients of themappingL which ensuretheexistenceof solutionsfor R andP’ satisfying(3.50)up to nth ordertermsin x andy. If theseconditionsaresatisfied,calculationof the R andP’ which satisfy (3.50)to orderncanbeachieved.Typically, not all thecoefficientsof R andP~arecompletelydeterminedsothat somechoice is available. Table 3.2 summarisesthe necessaryconditions on the second and third ordercoefficientsof theexpansionof L for themostcommonformsof its linear part. In case1 is includedthecasewhen the point is elliptic with complex-conjugateeigenvalues.As remarkedin section2.2, thiscomplex linear part differs from the real linear part on the right handsideof (3.22) but the respectiveexpansionscan be relatedto one anothervia a linear transformation.Note that for the typical linearpartsof an orientation-preservingmapping given in table3.2, the only possiblereversingsymmetriesare orientation-reversingin line with the assumptionon R above(an orientation-preservingreversiblemapping is either the productof two orientation-reversinginvolutions or two orientation-preservinginvolutions; the latter must havelinear part equalto — I (I is the identity matrix) abouta symmetricfixed point, which implies that dL = I at the point).

We illustrate the applicationof the local reversibility testswith the following example[QuispelandCape! 1989]:

y+ (K/21T) sin(2lTx’)L:x’=x+w(y)(modulol), y’= , . (3.52)1 — yh(x)


Table3.2Necessaryconditionsfor amappingL to be locally reversiblearounda symmetricfixed point. The conditions relate the

coefficientsof the expansionof L, given in (3.51),when its linear part hasoneof thetypical canonicalforms shown.

Linear part Case Secondorder Third order

(~~ A ~ ±1 1 no condition (A — 1)(Ag,2 + A’f2, —f,~g,,— 2f02g20)+f20f11(A

2 —2A’)

+g,1g02(2A

2 —

(~~),b ~‘ 0 2a’~ g,, + 2f20— 2bg,,= 0 if g2, = 0 then

3b2g,

0 — bg21 — 3bf,, + 2f20g02+f2~f,1— 3bf,,g,,= 0

2b”~ g2, = 0 6b2g,

0— 2bg,~— 6bf,0 + 2f20g02+ bf20g,1+ 6bf~0+ 2bg~1

— = 0

(~-~)‘ b ~0 3 no condition 12bg~+ 4g,1 + 12)’,, + 2g~1+ 2f20g1,+ 22bf20g2,+ 10b2g~,

+ 10f11g2,+ llbg20g,,+ 4g25g02+ 12f~,= 0

o) If linear part of involutionR is ~ ~ b) If linear part of involution R is (_1 ~).

For h(x’) = 0 and o(y) = y, this mappingreducesto the Chirikov—Taylororstandardmappinggivenby(1.25)which is area-preservingandreversible.Otherwise,if h(x’) is nonzeroandalsoodd, or w(y) isodd, thenit can be shownthat themappingis also reversible[Quispel1988]. It thenbelongsto a classof reversiblenon measure-preservingmappingsthat will be constructedin the following chapter(cf.classIV, section4.2). Weconsiderthecasewhenw and h arenotoddfunctionsbutcontainanevenpartvia the modifications

w(y) = Wodd(Y) + w2y2, h(x) = h

0 + hodd(x), (3.53)

where Wodd and hOdd arearbitraryodd functionsand w2 and h0 are constants.Wewantto knowwhether(3.52)with w and h givenby (3.53) is reversible,usingthemethodoflocal

reversibility. The fixed points(x0, y0) of themapping (3.52) haveto satisfy

w(y0)El, (KI2ir) sin(2~x0)= —y~h(x0).

For y0 =0, we have fixed pointsat x0 =0 and x0 = 1/2. Expanding w and h about0,

w(y)=w1y+w2y2+o.

3y3+0

5, h(x)=h0+h1x+03,

we find that the eigenvaluesof dL at (x0, y0) = (0,0) are determinedby

(1—A)2KAa

1, (3.54)

from which it follows that DetdL (0, 0) = 1. The eigenvaluesare different from those at (x0, y0) =

(1/2, 0) if Ku1 ~ 0, and are in generalalso different from the eigenvaluesat possiblefixed pointswith y0 � 0. Therefore,the origin is a symmetricfixed point if (3.52) is reversible.

JAG. Robertsand G.R.W.Quispel,Chaosand ti,ne-reversalsymmetry 105

To investigatethe local reversibility conditionsaroundthe origin, we expandthemapping to thirdorderaroundthis point. The result is

x’=x+u1y+w2y2+u

3y3,

L: y’= Kx+ Ily+(Kh0)xy+(Kw2+ h0[2)y

2+g3x

3+(3u1g3+ Kh1)x

2y (355)

+ (3u~g3+ h111 + K~I’)xy

2+ (Ku3 + u~g3+ Ku2h0 +

g3:——2ir2K/3, (2:=1+Kw

1, 11’:=h~+u1h1.

Applying to this expansionthe linear transformation

X=(A~—1)x—u1y, Y=(A—1)x—u1y, (3.56)

togetherwith (3.54), brings its linear part into the diagonal form (A, 0,0, A~).The transformedsecondorder and third order coefficients in the expansionare most easily found using an algebraicmanipulationprogram.Substitutioninto the third order local reversibilitycondition for case1 of table3.2 yields

—2u2h0(A— 1)A=0. (3.57)

From (3.57),it follows that themappingis not locally reversibleat third orderat theorigin if u2h0 ~ 0,i.e., when both u(y) and h(x’) contain an evenpart. Consequently,the mapping is not globallyreversible.

It is significant that the abovemapping (3.52) wasnot measure-preserving.The resultsof applyingthe test to a measure-preservingmapping follow from comparisonof table 3.2 with the necessaryconditionsfor local measurepreservationin table2.1. It is seenthat theconditionsfor cases1, 2 and 3of table 2.1 are identical to thosefor cases1, 2a and 3 of table 3.2. Therefore,to third orderthenecessaryconditionsfor local reversibility areimmediatelysatisfiedby measure-preservingmappingsinall thecasesof the linear partsshown. Consequently,this analysisto third ordercannotbe usedto findmeasure-preservingmappingsthat are not reversible. In fact, this equivalencebetweenlocal measurepreservationand local reversibi!ity extendsto arbitrary order for measure-preservingmappingswhenthe fixed pointhasa (normalised)Jacobianmatrix givenby thecasesof table3.2 (cf. RobertsandCapel[1992b]).This follows by observingthat aroundsuchfixed points in measure-preservingmappingsthereis a formal transformationto a normal form which is locally reversible(cf. Birkhoff [19201,Moser[1956,1968], Simó [19801).Consideringeachcaseof table3.2 in turn we have:

(i) Case 1. When the fixed point is hyperbolic (i.e., it is a saddlepoint) the normal form can bewritten

x’=xh(xy), y’=y/h(xy) (3.58)

with the function h uniquely determinedand h(0) = A. The mapping (3.58) can be verified to bereversiblewith symmetryS: x’ = y, y’ = x. Moser[1956]hasactuallyshownthat the transformationtothis form is morethanjust formalbecausea coordinatetransformationcanbefoundwhich convergesina neighbourhoodof the point. When the fixed point is elliptic, A e4i8 and A

1 = e-i8, with 0/2irirrational, thenormal form can be written


x’=xcosw—ysinu, y’=xsinw+ycosw (3.59)

with w = 0 + S(r) where S(r) = s1(x2+ y2) ~ is an infinite power series in r = x2 + y2. Equation

(3.59)hasthesameform asthe linear part of themapping— rotationaroundtheorigin. It is reversib!e,againwith symmetry5: x’ = y, y’ = x. In generalthe transformationtaking the mapping to theform(3.59) is only formal; it does not convergein any neighbourhoodof the fixed point. However, aconvergent(area-preserving)transformationcan alwaysbe found suchthat the transformedmappingagreeswith the normal form (3.59) to any desiredfinite order [Siegeland Moser 1971, section23].Consequently,the mappingaroundthe elliptic fixed point is locally reversibleto any finite order.

(ii) Case 2. When the fixed point is parabolicwith repeatedeigenvalue+ 1 and its JacobianhasnondiagonalJordannorma!form (i.e., b ~ 0), it follows from Simó [1980]that theexpansionaroundthepoint can be broughtto the form

x’=x+y+O,,~1, y’=y+F,,(x+y)+O,,~1, (3.60)

whereF,,(x + y) is a polynomialin (x + y) with termsofdegree2 to n, andn is anarbitraryinteger.Theexpansionto ordern is reversiblewith G: x’ = x + y, y’ = — y (cf. thegeneralisedstandardform oftable3.1) which correspondsto local reversibility case2a of table3.2.

(iii) Case 3. When the fixed point is parabolicwith repeatedeigenvalue—1 and b ~ 0, it followsfrom a similar procedureto that usedin case2 (cf. RobertsandCape![1992b])that thenormalform is

x’ = —x — y + O,,~, y’ = —y + F,,(x + y) + 0,,~, (3.61)

whereF,, is againa polynomialwith termsof degree2 to n, and n is an arbitrary integer.Theexpansionto ordern is reversible with G: x’ = —x — y, y’ = y.

It appearsthat, evenwhena normalform is not instantlyrecognisableaslocally reversibleasabove,normalformscanplay an importantrole in local testsfor reversibility, in thesamewaythat they do inlocal stability analysis (cf. Mira [1987]).For example, around a nonresonantelliptic fixed point(A q ~ 1), a real analytic mappingwhich is not necessarilyarea-or measure-preservingcan always beformally broughtto thecomplex form (cf. Moser [1968],Arrowsmithand Place[1990]),

L: x’ = Ax + ~ f,1x’y’, y’ = A

1y + ~ f,~x’y’, (3.62a,b)i+j=k�3 i+j=k�3

where A = &°, 0/2 ir is irrationai, i — j = 1, and the symbol * denotescomplex conjugation.Further-more, the reduction to the form (3.62) to a desiredfinite order can be achievedby an analytic(polynomial) transformation.Using thenonresonantnorma! form to developlocal reversibility condi-tions leadsto simp!er conditions. For example,local reversibility to third order leadsto the simplecondition

A~f21+ Af~’1= 0. (3.63)

This is the same at the condition for measure preservation of (3.62) to third order. Whentransformedbackvia the nonlineartransformationusedto obtain (3.62),the third ordercondition(3.63) becomesthat listed for case1 of table 3.2.

J.A.G. Robertsand G.R.W. Quispel,Chaosand time-reversalsymmetry 107

When we are interestedat a theoreticallevel in identifying whetherthereis a differenceat someorder betweenmeasurepreservationand reversibility, it makessenseto remove as many termsaspossiblefrom the mapping expansionand so usenormal forms. On the other hand,when applyingreversibilityteststo a specificmapping,ason (3.52) above,somework is neededto achievea normalform becauseit involvesusing a nonlineartransformation.It is practicallydesirableto havenecessaryreversibility conditions that only require a linear transformation before they can be used, as summarisedin table 3.2. It may still turn out (particularly in higher dimensions)that the derivationof theseconditionsis best doneby finding thecondition for a normal-formmappingand thentransformingthissimplerconditionby a nonlineartransformation.Formoreaboutan approachthatusesnormalformstoderivereversibility conditions,and leadsto reversiblenormalforms, seeLambet al. [1992](reversibleforms are also obtainedin, e.g., Sevryuk[1986]andMoser and Webster[1983]).

We concludethis sectionby asking when local reversibility is not consistentwith local measurepreservation.From the comparisonof tables3.2 and 2.1, it is seenthat at a symmetricfixed pointreversibility is usually consistentwith measurepreservationat thepoint, at leastto theorderworkedtohere.However,case2b of table3.2 is not in generalconsistentwith measurepreservation.In chapter7this casewill be relatedto a symmetry-breakingbifurcationthat producesan attractingfixed point andarepellingfixed point from thesymmetricfixed point.

3.4. Area-preservingmappingsthat are not reversible

The normal forms listed aboveindicate that the only possibilities for a mapping that is locallyarea-preserving(and orientation-preserving)arounda fixed point to not be locally reversiblearethatthe fixed point is parabolicwith linearisationgiven by the identity matrix I = (1, 0, 0, 1), or that thefixed point is a resonantelliptic fixed point with eigenvaluesA = e~’°where 0 = 2irp/q and p, q arecoprime. In this sectionwe follow RobertsandCapel[1992a,b] who havestudiedthe formercaseandshown that area-preservingmappingspossessinga fixed point where the linearisationis the identitymatrix are generallynot locally reversibleabout this point and so cannotbe globally reversible.Thecasewhere the linear partof the mappingexpansionequalsI wasnot consideredin the analysisof theprevioussectionasit is atypicalcomparedto thosein table3.2. This specialcaseis includedin the localreversibilityanalysisof Brown [1991].

Considera mappingL with fixed point p which hasbeenidentifiedasnecessarilysymmetricif L isreversible.Wetransferthepointto theorigin 0 andassumethat themappingexpansiontakestheform

L:x’=x+f2(x, y)+f3(x, y)+04, y’=y+g2(x,y)+g3(x,y)+04, (3.64a,b)

where the kth ordertermsof x’ are given by

~ f~,x’y’, (3.65)i+j =k

andsimilarly for g~(x,y) for k =2 and3, and the04 notationstandsfor 4th andhigherorder termsin xandy.

Area preservationof L imposesconditions on someof the coefficientsfq and gq~The JacobiandeterminantJ(x, y) of (3.64) takesthe form


~ ~)+o

Equatingthe first brackettedterm to 0 implies

g11=—2f20, g02=—~f11, (3.66a,b)

so that when (3.64) is area-preserving, the second order coefficients are specified by just{ f20’ f11~f02~g~0}.Similarly, equatingthesecondbrackettedterm to 0 implies

g21 = —3f30— 2f20g11+ 2g20f11, g12 = —f21 + 2f02g20— 2f20g02, (3.67a,b)

g03 = — ~(f12 + 2f11g02 — 2f02g11), (3.67c)

so that the third ordercoefficientsarespecifiedby {f30, f21, f12, f03, g30}.Considernow thecaseof a possible(orientation-reversing)symmetryG of (3.64)whoseexpansion

around0 haslinear part (1, 0, 0, —1),

G:x’=x+l2(x,y)+13(x,y)+04, y’= —y+m2(x, y)+m3(x, y)+O4. (3.68a,b)

It canbe shownthat if onewants (3.64),whenarea-preservingto third order,to havesucha reversing

symmetry, thennecessaryconditions on thecoefficientsin the expansionof L are

f20=f02=0, f~1—4f12=0. (3.69a,b)

Theconditions (3.69a,b) arederivedby finding theinverseL 1 of theexpansion(3.64) to third orderandimposinglocal reversibility with respectto G in (3.68) to third orderabouttheorigin in the form

L0G=G0L1. (3.70)

In doing this it is also necessaryto imposethat (3.68) is an involution to secondorder. The analysisrevealsthat condition (3.69a) is found to be necessaryfor local reversibility at secondorder andcondition(3.69b) is neededfor local reversibility at third order.

Conditions(3.69a,b) give additionalconstraintson the coefficientsof L, whenconsistentwith areapreservation,to be locally reversiblewith respectto just onepossibleinvolution (3.68). Nevertheless,theycan be usedto generateadditionalnecessaryconditionsfor local reversibility whenthe involutionhasthe expansion

x’ =px+ qy+12(x, y)+13(x, y)+04, (3.71a)

y’ = rx — py+ m2(x,y) + m3(x, y) + 04, (3.71b)

wherep, q and r are arbitraryrealnumberssatisfying

p2+qr=1. (3.72)

JAG. Robertsand G.R.W. Quispel,Chaosand time-reversalsymmetry 109

The linearpartof (3.71),togetherwith (3.72),is thegeneralform of d G abouta symmetricfixed pointfor 0 orientationreversing[cf. eq. (3.12)]. The analysisfor the simple orientation-reversinginvolution(3.68)can be usedto handlethe generalcasebecause:(i) the generallinear part of theorientation-reversinginvolution (3.71) can be reducedby a !inear transformationto that of (3.68), i.e., to(1,0, 0, —1); (ii) any linear transformationleavesthe form of (3.64) invariant, which is an essentialpoint and truepreciselybecauseits linearpart is theidentity; (iii) local reversibility is maintainedundercoordinatetransformation;and (iv) areapreservationto a certainorderis maintainedunder a linearcoordinatetransformation.Consequently,themappingL in (3.64) is reversiblewith respectto (3.71)for given { p, q, r} only if, after applying to L a linear transformationthat takes (p, q, r, —p) to(1,0, 0, —1), weobtainf

20 = f02 = 0 andf~1— 4f12 = 0 in the transformedmappingL. Theseconditionscanbe rewritten in termsof theoriginal coefficientsof themappingL.

Whatemergesfrom this analysisis that the linear partsof possibleorientation-reversinginvolutionsof (3.64), that is, thevaluesof {p, q, r} in (3.71),aredeterminedby the secondordercoefficientsofthe mapping expansionand are finite in number.A thoroughknowledgeof the possibleinvolutionlinear partsfor a given mappingallows its setof possibleinvolutions to benarroweddown (it canalsobe shown that there is always at least one possible linear part). A third order local reversibilitycondition for eachpossible orientation-reversinginvolution with a linear part allowed by the secondordercoefficientscanthenbe foundvia a linear transformationof thecoefficientsappearingin (3.69b).Becausethe linear part of (3.64) equals the identity matrix I, we need also derive third orderreversibility conditionsfor possiblereversingsymmetrieswhich areorientation-preserving,with linearpart equalto — I (cf. Brown [1991],RobertsandCape![1992b]).A mappingwhich satisfiesnoneof thenecessaryconditionsso obtainedfrom eachof its possibleinvolution linear parts canbe declarednotreversible.

This philosophycanbe illustratedin a direct way on the generalisedstandardmapping

L:x’=x+f(y), y’=y+g(x’). (3.73a,b)

As pointed out in section3.2, this mapping is area-preservingfor any functionsf andg, andknown tobereversiblewhenfor g is anodd function(cf. table3.1). Assumethat theorigin 0 is a fixed point withlinearisationequalto the identity matrix and that about0 the mapping expansiontakesthe form

=x +f,,y~+f,,~1y”~’+ Ofl+2, (3.74a)L:

y’=y+g,,x’~+g,,~1x’~~’+O,,÷2, n�2. (3.74b)

Of course,the typical casein (3.74)will befor n = 2 [whenit becomesa specialcaseof (3.64)] but theanalysisandresultsfor n >2 in this examplefollow in a similar fashion. Assumethat the nth ordercoefficientsf,, and g,, in (3.74) satisfy

f,,~0, g,,~0. (3.75)

Becauseof the particularform of (3.73) the expansion(3.74) is automaticallyarea-preservingto allorders,i.e., no conditionsneedbe imposedon thecoefficientsasin themoregeneralexpansion(3.64).

However,there areconditions on the coefficientsof (3.74) in order for it to be locally reversiblearoundtheorigin (assuming0 to be also fixed underpossiblereversingsymmetries).It is foundthat the

110 JAG. Robertsand G.R.W. Quispel,Chaosand time-reversalsymmetry

coefficientsf,, andg,, selectthepossibleinvolution linear parts,and thesecoefficientstogetherwith thecoefficientsf,, + and g,, +1 areinvolved in necessaryconditionsfor local reversibility at ordern + 1. Bychoosing the coefficients { f,,’ g,,~f~+~ g,, + 1 } in (3.74) appropriately,we can ensurethat none of thenecessaryconditions is satisfiedandso rule out local reversibility of themappingexpansion(3.74)withrespectto orientation-reversingsymmetryexpansionsaroundthe fixed point. It canbe shown that thechoicesof coefficientsthat rule out local reversibility of (3.74) and (3.75) with respectto orientation-reversingsymmetryexpansionssuffice to also exclude local reversibility with respectto orientation-preservingsymmetryexpansions.Consequently,global reversibility can be ruled out. This analysisissummarisedin the following result[RobertsandCape! 1992a]:

An area preserving mapping (3.73) with expansion (3.74) and (3.75) around a fixed point, andhaving no otherfixed point with linearizationequalto the identity matrix, is not reversibleif f~+1 ~ 0andg,,~1s~0and {f,,, g,,, f,,~1,g,,+1} do not satisfy thecondition

(fn)n+2(gn+

1Y” = —(g )~~2(f+

1Y~1. (3.76)

If n in (3.74) is even,in particularn = 2, the result can be extended;the mapping is also notreversibleif just one of f~+ 1 and g,, + 1 is nonzero [in which case (3.76) is immediatelyviolated]. Anexamplefrom theclassof nonreversiblearea-preservingmappingswith expansion(3.74)in the typicalcasen = 2 is provided by

M:x’=x—y2(1—y), y’=y+ Cx’2(1—x’). (3.77a,b)

Here C is an arbitrary parameter.This polynomialmappingobviouslyhastheexpansion(3.74)aboutthe fixed point at the origin with f

2 = —1, f3 = 1, g2 = —g3 = C. From the above result, it is notreversibleif C ~ 0 [cf.assumption(3.75)], and from (3.76), C~ 1. WhenC = 0, Mis reversiblebecauseg(x’) = 0 is odd. WhenC = 1, Mis reversiblewith reversinginvolution G: x’ = y, y’ = x because(3.73)is alwaysglobally reversiblewith this involution wheneverg(y) = —f( y). If we restrict (3.77) to C> 1,

we havea one-parameternonreversiblemappingof theplane.ThemappingM hasotherfixed pointsat(1, 0), (0, 1) and (1, 1). The fixed point at (1, 1) has a linearisationdependenton the mappingparameter C, seenby evaluatingthe traceof theJacobianmatrix which equals2 — Cso that thepoint iselliptic for 0< C<4. From alocal analysisaroundthe fixed point (1, 1) thelack of reversibility of M isless transparent.In fact when the fixed point (1, 1) is hyperbolic (C >4), the mapping is locallyreversibleto all ordersabout this point becauseof the normalform (3.58),eventhoughglobally themapping is not reversible.

Note that although (3.77) is definedover the entire xy plane, it is easy to createnonreversiblearea-preservingmappingsof the cylinder or the torusby choosing one or both of f( y) andg(x’) in(3.73) to be periodic functions. The way in which the lack of reversibility of mappingssuch as Mmanifestsitself in theirdynamicalpropertiesis yet to beinvestigated(cf. Robnik andBerry [1985,1986]for a discussionof the consequencesof breakingtime-reversalsymmetry in billiard dynamics).

The resultsof Robertsand Cape! [1992a,b] showthat quite generallyan area-preservingmappingpossessinga fixed point wherethe linearisationis the identitymatrix is not reversible.In thecaseof thegenera!isedstandardmapping (3.73),we can look atthe resultstateddirectly aboveeq. (3.76)in thefollowing way: if the functionsf and g in the generalisedstandardmapping (3.73) are suchthat the

JAG. Robertsand G.R.W.Quispel, Chaosand time-reversal symmetry 111

lowestorder in both of their expansionsarounda fixed point is cubic, then,whenwe addaneventermto both f and g, by virtue of nonzeroquadraticterms or nonzeroquartic terms the mapping isgenericallynot reversible. It would be interestingto know whetherthisconclusionextendsto the casewhen the expansionsof f and g havelinear parts and eventermsare presentby virtue of nonzeroquadraticterms.Rannou’smapping(3.47) is anexamplewherethis occurs.Aboutthe fixed pointat theorigin, we find

f(y)=y+y212+O(y4), g(x’)=—C(x’+ ~x’2)+O(x’3).

There is a relation betweenthe mappingexpansion(3.64) and the caseof a resonantelliptic fixedpoint, in the sensethatthe qth powerof the mappingexpansionaroundafixed point with eigenvalueAsatisfying A q = 1 has linear part (1,0, 0, 1). This would seemto suggesta programfor testing localreversibility at a q resonanceof an area preservingmapping: take the qth power of the mapping,analyseit as an example of (3.64) and check whetherthe local reversibility conditionsderivedformappingsof the form (3.64) are satisfied.The underlyingidea is that if L q can be shownnot to belocally reversible, thenit follows that L is not locally reversiblebecausethe property is preservedontaking powers.Preliminaryinvestigationsof this typehaverevealedthat L q for smallq doessatisfythelowest order reversibility conditionsderivedfor mappingslike (3.64).

This can alsobe seenfrom a direct approachbasedupon nonlinearnormal forms. At a resonanceareal analytic mapping,not necessarilyarea-or measure-preserving,can alwaysbeformally broughtviaa nonlineartransformationto the complexform (3.62)wherenow i — j = 1 (mod q). Furthermore,thereduction to this form to a desired finite order can be achieved by an analytic (polynomial)transformation.The fact that local measurepreservationand local reversibility are maintainedunderthe reductionto normal form can be exploited by studyingthe conditionsunder which the reducedmappingis measure-preservingandreversible.It is not hardto show that for smallq (i.e., q = 3, 4), theexpansionof L in this caseis locally reversibleat the first few ordersof theexpansionwhenit is locallymeasure-preserving,andvice versa.Furthersystematicinvestigationof the normalform is requiredtodetermineif local reversibility existsto all ordersarounda fixed point of an area-preservingmappingata resonance.

4. Non measure-preservingreversible mappings with asynunetric fixed points

This chapterintroducessomelargeclassesof non measure-preservingreversiblemappings.Thebasicfeaturesof someexamplesfrom theseclasseswill be discussedhereandtheywill be studiedin moredepthin chapters5 and 6. It will be shownthat their dynamicsincludeschaoticmotion, attractorsandrepellers, as well as motion on invariant curves. It is not sufficient to make a mapping nonarea-preservingin order to introduceattractorsor repellers.For example,a reversiblemappingmayhavea Jacobiandeterminantthat varies throughoutthe planebut it mayin fact be conjugateto anarea-preservingreversible mapping. Certainly this class of reversiblemeasure-preservingmappingscannothaveattractorsor repellers.*)

What is necessary,therefore,in order to haveattractorsand repellersis that the mappingbe non

*) This may explainthe statementsin Belobrov et al. [1984]that reversiblemappingscannot haveattractors— it seemsthat the reversible

mappingsstudied therecan be transformedto area-preservingforms.

112 l.A.G. Robertsand G.R.W. Quispel, Chaosand time-reversalsymmetry

measure-preserving.For the orientation-preservingreversible mappingsdiscussedbelow, it will beshown analyticallythat they can possessfixed pointsat which theJacobiandeterminantis notequaltoone. Thesepoints arenecessarilyasymmetricfrom the discussionof section3.1, and furthermorefromsection2.2 it follows that thesemappingscannotbemeasure-preserving.In fact, often theseasymmetricfixed points in one-parameterfamilies of reversiblemappingsareattractorsor repellersin somerangeof the mappingparameter.This can be shown explicitly using linear stability analysis.In one of thereversiblemappingsbelow therealso appearasymmetricfixed pointsatwhich I = 1. Computerstudiessuggestthat thesepointscanalso be attractingor repellingin a rangeof the mappingparameter.Thiscan be confirmed by higher orderstability analysis.

Reversiblemappingswith attractorsand repellersarosein thework of Politi et al. [1985,1986a,b]and Bullett [1988,1991]. In the former casea specificexampleof a nonconservativereversibleflowarising from a physical examplewas studied,and the reversible mappingarose from a surfaceofsection.In the latter case,a specialclassof mappingsof part of the complex planewas studiedandattractingandrepellingperiodicorbits arosein somecases.The discussionbelow will indicatehow toexplicitly and systematicallyconstruct large families of reversible mappings with attractorsandrepellers.Thesefamilies allow a study of reversiblesystemswithouttheneedto takesectionsof flows,andareconvenientfor creatingreversiblenon measure-preservingperturbationsof reversibleconserva-tive mappings.

4.1. Theory

The constructionof reversiblemappingswith attracting/repellingfixed pointsrequirestheconstruc-tion of mappings which possess(i) asymmetricfixed points; and (ii) compriseat least one nonarea-preservinginvolution. Recall from (3.16) that a point x0 is a fixed point of a reversiblemappingL Ho G if and only if

Hx0=Gx0. (4.1)

If Gx0 = x0 (andhenceHx0 = x0) thenthe fixed point is symmetric,hasI = 1 (whenL is orientation-preserving)and cannotbe an attractoror repeller. If for amappingtheredoesexista point satisfying(4.1) with Gx0~ x0, thenthe fixed pointx0 is asymmetricand thepoint Gx0 is anotherfixed point of L.

It appearsthat the best “first principles” way of creating non area-preservinginvolutions is byconjugatingarea-preservingones.This methodwasusedby Pinnow[1986].The involution thuscreatedcan be usedto makeanon measure-preservingreversiblemapping(recall that althoughany involutionis necessarilymeasure-preserving,thecompositionof two measure-preservingmappingsneednot be).To usePinnow’s method, take an area-preservinginvolution V and conjugateit with an arbitraryinvertible coordinatetransformationW,

H := W0V0W1. (4.2)

The resulting mapping H is an involution but it is in general not area-preservingif W is not

area-preservingsince

DetdH(x)=±DetdW1(x) (4.3)DetdW (Hx)

J.A.G.Robertsand G.R.W.Quispel,Chaos andtime-reversalsymmetry 113

The + or — sign is takenaccordinglyas V (and thereforeH) is orientation-preservingor orientation-reversing.The conjugacyof H to Vensuresthat if x~is a fixed pointof V thenWx0 is a fixed point ofH.Hencethe symmetry line of H is the image underW of the symmetryline of V.

When H is composedwith anotherinvolutionG we obtain a reversiblemappingthat is in generalnotareapreserving,*)for example,

L=H0G=W0VoW’oG. (4.4)

If G is area-preservingas well as V and they are both orientation-reversing(the casewe usually

consider), thenthe Jacobiandeterminantof L is

DetdW’(Gx) (4.5)DetdW

1(Lx)

so that at a fixed point x~it is

J(x ~ DetdW1(Gx0). (4.6)0 DetdW’(x0)

If x0 is asymmetricso that x0 ~ Gx0, then(4.6) showsthat in general1(x0)~ 1 which meansthat L isnot measure-preserving.

The abovemethodcan be illustratedwith a simple example.Take

V:x’=y, y’=x, (4.7)

W:x’=x[1+(y—1)2], y’=y. (4.8)

ThetransformationW would appearto be oneof thesimplestexamplesof aninvertible transformationwhich is not area-preserving.With thesechoicesweobtain for thenon area-preservinginvolution H in(4.2),

Hi:xl=y[1+(y?_1)2], y’=x/[1+(y—1)2J, (4.9)

which we composewith the simple one-parameterarea-preservinginvolution

G1:x’=x, y’=C—y. (4.10)

The resultH1 o G1 is the reversiblemapping

L1: x’ = (C — y)[l + (y’ — 1)21, y’ = x/[1 + (C — y — 1)2]. (4.11)

One finds that for a substantialrangeof C this mappinghasa symmetricfixed point and apair of

s) As far as we are awarePinnowdid not createreversiblemappingswith asymmetricfixed pointsat which theJacobiandeterminantwas not

equal to 1.

114 J.A.G.Robertsand G.R.W.Quispel, Chaosand time-reversalsymmetry

asymmetricfixed points, e.g., at C = 3 theseare readily calculated:the symmetric fixed point is at(15/8,3/2) and the asymmetricpair at (2, 1) and (2,2). Furthermorethe Jacobiandeterminantof(4.11) can be foundusing (4.8), to readily find W1, followed by (4.5),

J1(x) = [1+ (y’ — 1)2]/[1 + (C—y — 1)2]. (4.12)

So at the asymmetric fixed points when C = 3, we find 11 = 0.5 and J1 =2 respectively.Furthercalculationof the linearisationat thesepointsrevealsthat theyarein fact, respectively,attractingandrepellingfixed points.Fuller detailsfor the mappingL1 will begiven in section4.3 (seeexample1, table4.1). In fig. 4.la,b apart of the phaseportraitof thismappingis shown.Figure4. ic showsthe variationof the Jacobiandeterminantover a portion of the plane.

4.44 2135

• ~- -----.~

~

((~/

___________________________-1-57 ~ ___—139 470 ‘—.-..- O•735

(a) 12065 (b) 2~2065

Fig. 4.1. (a) Phaseportrait of themappingL, in (4.11)whenC = 2.87 (seealsoexample1 of table4.1 in section4.3).For clarity thex andyaxesarenotdrawn.At thecentreof thepictureis asymmetricfixed point whichliesat the intersectionof thesymmetryline of G, which isy = 1.435,andthesymmetryline of H, whichis themonotonicincreasingcurveshown.The symmetricfixed point is hyperbolic(a saddle),andbelowandaboveitare two asymmetricfixed points that form a spiral attracting/repellingpair. A trajectory is shown that startsclose to the repellerabove thesymmetryline of G andmovesdownover the line to spiral into theattractingfixed point. The attractorandrepellerandthehyperbolicsymmetricfixed point appearto be enclosedby invariant curves.Some elliptic symmetriccycles andtheir associatedislands are alsoshown: a symmetriceight-cycle and a symmetricseven-cyclewhich lie, respectively,inside andoutside, oneof thecurves,andanouter symmetricsix-cyclewhich issurroundedby largeislands.Thethreeislandscut off by theborderof thepicture arepart of the island chainsurroundingasymmetricfive-cycle.Onechaoticorbit is alsoshown. (b) Enlargementaroundthe centreof fig. 4.la. The coordinatesof the hyperbolicsymmetricfixed point at thecentre are (1.7065.. . , 1.435). The attracting and repelling asymmetricfixed points below and above the horizontal symmetry line have,respectively,coordinates(1.74,1.1916...)and (1.74,1.6783.. .). We have addedtwo trajectoriesnot shown in a. The first appearsto bequasiperiodicandproducesthe innermostapparentlyclosedcurve.Thesecondadditional trajectoryis thatof thepoint markedPwhichstartscloseto this curve andmovesanti-clockwisearoundit towardsthehyperbolicsymmetricfixed point, beforechangingdirection andspiralling in towardsthe attractor.Therecan be no invariantclosedcurveswhich enclosethesymmetricand two asymmetricfixed points andwhichhave P in theirexterior.(c) Plotof theJacobiandeterminant(4.12)of themappingL1 in (4.11) over thepartof theplaneshownin b. Thetoptwo picturesshowthesurfaceobtainedby plotting J(x, y) whenviewed from two different perspectives.OvertheregionshownJ(x, y) variesfrom approximately0.5to 2.5, mostly increasingwith increasingx andincreasingy. In thebottompicture weshowonly thepartof thesurfacewhere1> 1. The repellingfixed point whichhasI = 1.408...is containedin thepart of theplanebeneaththispart of thesurface.Theboundaryof thispartof thesurfaceisthecontourwith I = 1, on which lies thesymmetricfixed point. In reality this is asmoothcurvebut its jaggedappearancehereis anartefactof thegraphicspackage.

J.A.G.Robertsand G.R.W.Quispel, Chaosand time-reversalsymmetry 115

2.52

1.5 21

0.51.2 1.5

x 1.41. 2 2. /

2.52

1.2 1.5

X 1.4 0.52

2.5

.2~cç ~1.5

(c)Fig. 4.1 (cont.).

It appearsthat the method just describedis the most systematicway of creatingreversiblenonmeasure-preservingmappings.It is also very generalbecausemany invertible transformationscanbeusedfor W. In section3.2 it was shownhowinvolutionscould be derivedfrom symmetricsecondorderdifference equations.Some of these involutions were not area-preserving.It turns out that theinvolutions createdby this method are in fact further examplesof the Pinnow method becausetransformationscan be found that reduce them to area-preservingform. Thus, for example, the

116 l.A.G. Robertsand G.R.W.Quispel, Chaosand time-reversalsymmetry

involution (3.45) canbe written WoVoW1 with V: x’ = x, y’ = —y, and

W~:x’ = x, y’ = y3f(x)+ y2m(x)+ yg(x)+ h(x)/2. (4.13)

The transformationw1 is invertiblebecauseof the assumptionm2— 3fg<0.

4.2. Four classesofnon measure-preservingreversiblemappings

In this sectionwe give fourclassesof non measure-preservingreversiblemappingscreatedin thewayjust outlined. Theseclassesvary in the simplicity and analyticity of their mappingsand the easeoffinding fixed points, etc. Eachclass involvesarbitrary functionswhich, with appropriatecare,can bechosento guaranteethat themappingshaveasymmetricfixed pointswheretheJacobiandeterminantisoftennotequalto 1. This will beillustratedin thenext sectionby creatingexampleswith simple choicesfor these functions. Here we list the basic form of the mapping in each class, its componentorientation-reversinginvolutions, at least one of which is not area-preserving,and the mapping’sJacobiandeterminant.From this informationsuchdetails assymmetrylines, fixed points[cf.eq. (4.1)1andthe Jacobiandeterminantat fixed points arereadily obtained.The readermay preferto skip overthis sectionand proceedstraightto thespecific examplesof section4.3, and in particularto table4.1,where mostof thesalient featuresof somemodel mappingshavebeencalculated.This sectioncanbeusedfor referenceon the class to which eachexamplebelongs.

ClassesI and II below are analyticmappingsof theplanethat are useful for perturbingreversiblearea-preservingmappingswritten in McMillan form (cf. table 3.1), which for easeof referencewerewrite hereas

L~:x’=y, y’=—x+2h(y), (4.14)

with its involutions

HM:x’=x, y’=—y+2h(x); G~:x’=y, y’=x. (4.15)

Class III also consistsof analyticmappingsbut thesemappingsinvolve takingcuberoots to get theimage point (x’, y’) from (x, y). ClassIV mappingshavesingularity lines in theplanewheretheyarenot defined.ClassesIII and IV mappingsare usefulfor perturbingreversiblearea-preservingmappingswhenthey arein generalisedstandardform~

L~:x’=f(y)+x, y’=g(x’)+y. (4.16)

In the casethat f is odd, the involutions are

H~:x’=x, y’=g(x)—y; G~:x’f(y)+x, y’=—y, fodd. (4.17)

L~in general can possessasymmetric fixed points. However since L~is area-preserving,theseasymmetricfixed pointscannotbe attractors.In themappingsof classIII and classIV below, one orboth of the involutions in (4.17) is replacedwith a non area-preservinginvolution which allowsasymmetricfixed points to be attractingor repelling. ClassesIII and IV can also be usedto perturb

J.A.G. Robertsand G.R.W.Quispel, Chaos and time-reversalsymmetry 117

mappingsin McMillan form (4.14) after first convertingthem to generalisedstandardform using the

transformationx—~x,y—+x — y.4.2.1. ClassI

This classof mappingstakesthe McMillan mapping (4.14) and (4.15) and conjugatesits secondinvolution to give

LI=HMoWoGMoW~. (4.18)

The transformationchosenfor W is

W:x’=xf(y), y’=y. (4.19)

The functionf( y) is takenstrictly positive or negativeso that the inversew1 is everywheredefined.

The mappingL1 is thereforethe productof the involutionsH1 HM and

GI=_WOGMOW’:x’=yf(y’), y’=x/f(y). (4.20)

Observethat the transformationW in (4.19)generalisestheW in (4.8) andthat, sinceGM is equalto Vin (4.7), the involution G1 generalisesthe involution H1 in (4.9). The explicit form of L1 = H1° G1 is

L1:x’ =yf(x/f(y)), y’ = —x/f(y)+2h(x’). (4.21a,b)

Its Jacobiandeterminantis given by the right-handside of (4.3), with the + sign andH—~G1, because

in (4.18) the conjugatedinvolution is after the area-preservingone,J1(x, y) =f(—y’ + 2h(x’))/f(y), (4.22)

with x’(x, y) andy’(x, y) obtainedfrom (4.21). Whenf~1, the transformationsW and W1 become

the identity transformationand the mappingL~is the area-preservingMcMillan mapping LM.

4.2.2. Class IIThis class is againderivedfrom the McMillan mapping (4.14) and (4.15) but in this casethe first

involution is conjugatedto give

LII=WOHM0W’oGM. (4.23)

Herewe take

W: x’ =[x— q(y’)J/p(y’), y’ =y—r(x), (4.24)

p(y) >0, even; q(y) odd; r(x) not odd, r(0) = 0. (4.25)

This transformationhasJacobiandeterminantl/p(y’). From (4.23), togetherwith (4.15) and (4.24),the explicit form for L

11 can be derived,


L11:x’=[j..t.—q(y’)]/p(y’), y’=—x+2h(~)—2r(p~), (4.26a,b)

~=yp(x)+q(x). (4.27)

Herethevariable~.t is an auxiliaryvariableintroducedto makeL11 neaterto write down.To calculate(x’, y’) onefirst evaluates~ andsubstitutesit andx into (4.26b)to find y’. The valuesof jL andy’ arethenusedin (4.26a) in order to find x’.

The reversiblemapping (4.26) is the productof the involutions

H11raWoH~oW’:x’=[P_q(y’)J/p(y’), y’=—y+2h(~)—2r(v), (4.28)

v=xp(y)+q(y), (4.29)

where ii is anotherauxiliaryvariableand G11 GM. The Jacobian determinant of L11 follows from (4.5)andthe Jacobiandeterminantp(y) of w 1,

J11(x, y) =p(x)/p(y’), (4.30)

with y ‘(x, y) given by (4.26b). It is alwaysdefinedand positive becausep(y) >0.The reasonfor theparticularchoiceof W in (4.24) and(4.25) is that it ensuresthatL11 hasthepair

of fixed points

(x0, y0) = {(x0, —x0), (—x0, x0)}, x0p(x0) — q(x0) = 0. (4.31)

This holdsprovidedh(0) =0, i.e., providedthat theoriginal McMillan mappinghastheorigin asa fixedpoint [cf. (4.14)],which is equivalentto it possessingany fixed point. The abovepair of fixed pointsofL11 are in generalasymmetricbecausewhenx0 ~ 0 they do not lie on the symmetryline y = x of G11.Moreover L11 doesnot havethe symmetryline y = —x becauseit cannotbe odd if r(x) is not an oddfunction.From (4.30),theJacobiandeterminant~H at thepair of fixed points is equalto + 1 becausepis an evenfunction. Despitethis, in the casesthat we haveinvestigatednumerically(e.g., seeexample2, section4.3) we find that thesepoints form a spirally attracting/repellingpair whentheyareelliptic.This suggeststhat they aregenuinelyasymmetric.This is confirmedby higher orderstability analysisaroundthe pQintswhich is discussedin chapter6. Note alsothat by construction[cf.(4.25)] the origin iscertainly alwaysa symmetricfixed pointof L11. Othersymmetricfixed pointsmaycertainlyexistbut areharderto find analytically.

4.2.3. Class IIIThe third classof non measure-preservingreversiblemappingstakesthe form

L111:x’=p(y)+x, (y’3—y3)q(x’)+(y’2+y2)r(x’)+(y’—y)s(x’)+t(x’)=0. (4.32)

In (4.32),t is a completelyarbitrary functionandp is anarbitraryodd function.Thefunctionsq, r andsmustsatisfy

r2(u) — 3q(v)s(u)<0, all v, (4.33)

JAG. Robertsand G.R.W.Quispel,Chaos and time-reversalsymmetry 119

so that q and s arenecessarilyboth everywherepositive or both everywherenegative.With thesemildrestrictions,(4.32) is reversibleandassignsauniqueimagepoint (x’, y’) to a givenpoint (x, y). This isbecausethe left handside of the secondequationin (4.32) is a monotoniccubic function for y’ as afunctionofx’ andy [asa resultof (4.33)] andhencehasonly one realzero.The explicit expressionforthis unique value of y’ follows from standardformulae for the root of a cubic (cf. AbramowitzandStegun [1972,section3.8.21).Theseformulaeareneededfor computationalwork with the mapping.

The non area-preservinginvolution H~11in this classof mappingsis

H111:x’=x, (y’3+y3)q(x)+(y’2+y2)r(x)+(y’+y)s(x)+t(x)=0. (4.34)

This involution is obtainedfrom a secondorder symmetric difference equation[cf. (3.45)] but, asremarkedabove,is conjugateto a simple area-preservinginvolution via (4.13). The area-preservinginvolution G

111 is of the form of G~in (4.17) with f—÷p.The mapping L111 = H111 ° G111 is in generalanon area-preservingmappingwith

— 3y2q(x’)—2yr(x’)+s(x’) 435

111 X~~ — 3y’2q(x’) + 2y’r(x’) + s(x’)

In (4.35),x’(x, y) andy’(x, y) arecalculatedfrom (4.32).The mappingL111 canbe usedfor reversible

non area-preservingperturbationsof the generalisedstandardmapping (4.16). It reducesto anarea-preservingmappingof this form whenq = r 0. If the functionsp, q, r, s and t are analyticandsatisfy the conditionsabove,then L111 is an analytic diffeomorphism.

4.2.4. ClassIVIn this classwe considermappingsgiven by

L f1(y)+x , g~(x’)+y 436 b— 1—xf3(y)’ ~ — 1—yg3(x’)’ . a,

where g1 and g3 are completelyarbitrary functions and f1 and f3 are arbitrary odd functions. Thefunctionsf1, f3, g1 and g3 canbe chosento be analytic(e.g.,polynomial or rational) andto dependononeor moremappingparameters.

The mappingL1~is theproductof the involutions

H1~:x’ = x, y’ = ~ (4.37a)

G1~:x’ = , y’ = —y. (4.37b)

The involution H1~is of the form (3.42) with f2 in (3.42) taken equal to + 1 and the obviousreplacements.G1~is of the form (3.43) with x ~ y andf2 theretakenequalto —1. Whenf3 = g3 0,H1~and G1v becomeequal to H~and G~in (4.17) and this classreducesto the area-preservinggeneralisedstandardmapping (4.16). However, H1~and G1~arenot in generalarea-preserving.TheJacobiandeterminantof Liv is

120 J.A.G. Robertsand G.R.W. Quispel, Chaosandtime-reversalsym,netry

Table 4.1Someexamplesof non measure-preservingreversiblemappingsL= Ho G

InvolutionsMapping L[Jacobian H GdeterminantI] [Symmetryline] [Symmetryline]

Examplel x’=(C—y)(y’2—2y’+2) x’y(y2—2y’+2) x’x

[Class!]

y’= 1+(C—y—1)2 y’ 1+(;_1)2 y’C—y

[1 1+(c_y_i)2] [x=y(1+(y-1)2)~ [y=C/2]

Example2 ,~—2e2y’3 ,_ v—2e2y”[ClasslI] X — 1+e2y’2 X — 1+e2y’2 x y

y’=—x+2C~+2~2 y’—y+2Cv+2v2 y’=x(j.~=y+e2x2y+2e2x3) (vx+e2y2x+2s2y3)

[ 1+e2x2 1 [ —(4e~y~+C)±\J4y+C21V 1+e~y’~J V 2(1+e2y2) [y=x]

Example3 = —y + s2y’+ x x’ x x’ = —y+ e2y3+ x[ClassIII] (y3 — y3)s2— (y’2+ y2)e (y3 + y3)e2— (y’2 + y2)e y’ = —y

+(y’-y)—2x’(1-C—x’)=0 d) +(y’+y)—2x(1-C—x)=0

[1= ~ [e2y’—ey2+y+Cx+x2-x——0] [y=0]

Example 4 — —y/(1 + e2y2)+ x o , — — —y/(1 + e2y2)+ x[ClasslV] X — 1—xy’/(I+y21e2) x —x X — 1—xy3I(1+y2/e2)

y’=2x’(l— C—x)+y y’=2x(1—C—x)—y y’= —y

1~(1+y2/e2)(1+e2y2+y2Ie2)~ — ~—~— =0L (1+s2y2)(1+y2/e2—xy’)2J [y—x( x)] [y ]

o) Asymmetricfixed pointsexist only when id > 2V~andbifurcatefrom symmetricfixed point when C = ±2v’~.b) Anothersymmetricfixed pointexists asshownin fig. 4.2a.o) Positionof asymmetricfixed pointslisted is independentof C; linearstability of asymmetricfixed pointslisted is independentof s. Thepoint

(1 /~,—1Is) hasI = 1 and for s>0 is apparentlyspirally attracting when it is elliptic. Note that otherasymmetricfixed points can be foundcomputationally,cf. chapter6, section6.1.1.

— [1 + g1(x’)g3(x’)J[1 + f1(y)f3(y)] 4 38

J~~(x,~ - [1- yg3(x’)]2[1 - xf

3(y)]2 ‘ (. )

with x’(x, y) calculatedfrom (4.36a).Clearly it varies throughouttheplane.An obvious featureof themappingsof this classis that they possesssingularities,i.e., curvesin the

J.A.G.Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry 121

Table 4.1 (cont.).

Symmetricfixed points Asymmetricfixed points s(I = 1 at thesepoints) (given in pairs)

(IC[1 + (1— jC)2], jC); (i) (i’ y,) (2(C—2), j(C ±

C(C-2) C±~/~~ (4-C)(C±V~i)trace 1 + (1— IC)2 2—V’~�traces2+~ = c—+v~i, trace,= 2(C—2)

elliptic for —2V~< C<2V~ (x, y_) and(x, y +) areattractor/repellerpairfor all C> 2V~o)

(x, y~)and(x, y) areattractor/repellerpairfor all C< —2V~a)

(0,0)~ trace= 2C (j)C) (±1/c,~1 Is); I = 1; elliptic for <C <elliptic for —1 <C <1 1.1

/1—C+~(1—C)2+4/s 1\ —1 1(0,0); trace=2C (i)’~ ~ 2

elliptic for —1< C< 1attractor/repellerpair for 0<(1 — C)2 +4/s <16

f1—C—~(1—C)2+4Is 1\ —1 1(1 — C, 0); trace= 2(2— C) (~~ 2 , ±—); J = ~,3 for y= — , — 1.0

elliptic for 1< C <3this pair of pointsalwayssaddlepoints

(0,0); trace=2C (i) (1— C, ±y1),y, >0: s’(1 — C)

2y1’ + (52(1 — C)

2 — 1)y1

2 — ~20

elliptic for —1 < C< 1 (thereisa positive solutiony1

2VC~ 1, s~ 0)(1— C,0); trace=2(2—C) (1— C, ±y,)often attractor/repellerpair for fixed sand 0.02

elliptic for 1<C<3 somerangeof C, e.g.,for s=0.02,(1— C, —y1)

is attractingfor — 12.943. . .< C< 1

d) To calculatey’ in this mappingusethe formulafor the root of a cubic polynomial.

°~If s<0,asymmetricfixed points do not exist for all C; two pairsarise by tangentbifurcation when (1 — C)2 +4/s= 0.

° Singularity line of mapping: x = 1 /y3 + 1 /e2y. Note that 1 +I,(y)f3(y) >0 Vy, s so unique inversefor everypoint, cf. (4.39).

~ Symmetrylines arethe sameasin Hén~nmap.

planewherethedenominatorsappearingin themappingsand their derivativesvanish.Theappearanceof thesingularitiesin bOth themappingL1~= H1~° G1~andits inverseL ~j= G1~ °~ canbe regardedas due to thepresenceof singularitiesin the componentinvolutions H1~and~ The effect of suchsingularitiesis that somepoints in theplanepossessonly backwardsorbits while otherspossessonlyforward orbits. It is also possiblefor a curve of points to all havethe sameimage point underL1~.


Howeverthis lossof injectivity can be removedby ensuringthat f1, f3, g1 and g3 satisfy

1+g1(v)g3(v)>0, 1+f1(v)f3(u)>0, all v. (4.39)

This conditionalso ensuresthat thesymmetryline of Hiv, which is typically double-valuedif g3(x)~ 0,extendsthroughoutthe planeand doesnot terminate.Note that somearea-preservingmappingshavebeenstudiedwhich havesingularities(e.g.,Devaney[1981],McMillan [1971])ornon-uniquebackwardsorbits associatedwith points(cf. GumowskiandMira [1980,chapter3.9J).Finally, like the generalizedstandardmappings,if g1 and g3 in (4.36) are odd insteadof (or togetherwith) f1 and f3, then themappingL1~can be written as the product of — H1~and — ~ The basic propertiesof this formresemblethosedescribedabove.

4.3. Examples

We now presentsomespecific examplesof mappingsin the four classesdiscussedabove. These

examplesand some of their details are listed in table 4.1. They illustrate specifically many of the

_____________________1~0 —087

Y

>‘•.•. ~ 1~0

-10 0~87 095

(a) (b)

Fig. 4.2. (a) Phaseportrait of the mapping example 2 of table 4.1when C = 0.3. The line y= x andthecurvepassingthroughtheorigin arethesymmetrylinesof G andH respectively.The origin is an elliptic symmetricfixed point and so is surroundedby invariantKAM curves.Theotherpoint of intersectionof thesymmetry lines is ahyperbolicsymmetricfixed point. Theasymmetricfixed pointat (—0.909... ,0.909.. .) is observedto be spirally repelling. Its reflection is theasymmetricfixed point at (0.909.. . , —0.909...) whichis observedto be spirally attractingasshowninb. (b) Enlargementof thebottomright handcornerof ashowingthespirally attractingfixed point(0.909. . . , —0.909. . .) at thecentre.Also shownis aspirally attractingfive-cycle. (c) Plotof theJacobiandeterminantof example2 overthepartof theplaneshownin a. Overthis regionJ(x, y)rangesfrom closeto 0 to greaterthan2.Thetop two picturesshowthesurfaceobtainedby plottingtheJacobiandeterminantwhenviewedfrom twodifferentperspectives.Thesurfaceis humpedin thegeneraly= —xdirectionandfalls awaysharplyfrom this direction. Thebottompictureshowsthepartof thesurfacewith 1>1. In this examplethesymmetricandtwo asymmetricfixed pointsall haveI = 1 andso lie in theplanebeneaththeboundaryof thispartof thesurface.Notethatquite nearto thespirally attractingfixed pointat(0.909.. . , —0.909. . .) areregionswhereJ(x,y) isgreatest.


(c)Fig. 4.2 (cont.).

propertiesof a non measure-preservingreversiblemapping that were previouslydiscussedin generalterms in section3.1. Example1 is the simple exampleintroducedin section4.1. It dependson oneparameterC. It is composedof two involutionsof thetypepresentedin classI of theprevioussectionexcept that it has the conjugatedinvolution in front of the area-preservingone. Neverthelessweconsiderit as a very simple mappingof this type. Examples2—4 dependon two parametersC ande.The parametersareusedin thefollowing way: wefix e andstudythe resultingone-parameterfamily ofmappingsparametrisedby C. The valueof e adoptedfor numericalstudy of eachmappingis listed in

124 I.A.G. Robertsand G.R.W. Quispel, Chaosandtime-reversalsymmetry

table 4.1. It will be shown in chapters5 and 6 that for thesefixed values of r, the one-parameterfamilies of mappings exhibit both qualitatively and quantitatively conservativeand dissipative be-haviour.

Unlessotherwisenoted,whenwe refer to examples2—4 we meantheexampleswith thevaluesof egiven in table4.1. It is worth pointing outhoweverthat thereis nothingparticularlyspecialaboutthevaluesof g chosen.As indicatedin table4.1, thesemappingsare reversibleand non area-preservingwith asymmetricfixed points for almostall valuesof C and ~. Thelinear stability of thesymmetricfixedpoints is constructedso that it is independentof e (becauseJ = 1 at thesefixed points, the linearstability is completely determinedby the trace of the Jacobianmatrix). The linear stability of theasymmetricfixed points usually dependson ~ and C, and for eachs thereis often a rangeof theparameterC (which usuallydependson s) for which thepointsareattractingor repelling. [Therangeofparameterfor which a fixed point is attractingcan bededucedanalyticallyin manycasesby using eqs.(2.lOa,b) with n = 1.] Of course(from section3.1) if thestability of onepointof an asymmetricpairoffixed points is known, then the stability of the otherpoint is also known becauseof their reciprocaleigenvalues.

The parametere in examples2—4 is in fact a perturbationparameterbecausetheseexamplesarederived from the area-preservingHénon mapping. The area-preservingHénon mapping has beenchosenas the basis for examples2—4 becauseof its pre-eminenceas a numerically and analyticallystudiedmappingof theplane.Theseexamplesillustrate howthemappingsof classesII, III andIV canbe usedto introducedissipativeand expansivedynamicalfeaturesinto conservativesystems.

Example2 (classII) derivesfrom thearea-preservingHénonmappingin McMillan form, (4.14)withh( y) = Cy + y2. Thesymmetrylines of thearea-preservingHénonmappingin this form arey = Cx +

andy = x. The mappinghastwo fixed pointswhich areboth symmetric.Thepoint (0,0)is elliptic when—1 < C < 1 and thepoint (1 — C, 1 — C) is elliptic when1 < C <3. The mappingof example2 reducesto this area-preservingmappingwhens = 0. When e ~ 0, example2 hasasymmetricfixed points withJacobiandeterminantequalto unity, which arebornat infinity andmovein to theorigin alongthe liney = —x with increasings. Numericallythey arefoundto be a spiral attracting/repellingpair whentheirlinearisation is elliptic. In figs. 4.2a—c we show the phaseportrait of example2 and its Jacobiandeterminant.Figure 4.2ashouldbe contrastedwith thephaseportrait of theHénonmappingshown infig. 3.2. [If in example2 one appliesthe scalingtransformationx—+x/sand y—+y/s, the asymmetricfixed points areat (1, —1) and(—1, 1) for all s and C; figs. 1.3 and3.1 showthis transformedmappingwhens=20and C0.3.]

Examples3 and 4 are derivedfrom the area-preservingHénonmapping in generalisedstandardform, (4.16) with f(y) = —y andg(x’) = 2x’(l — C — x’). In this form thesymmetrylines of the Hénonmappingarey = x(1 — C — x) andy = 0 and its (symmetric)fixed pointsare(0,0) and(1 — C, 0). Theirstability is thesameasin the McMillan form. Examples3 and 4 possessthesamesymmetricfixed pointswith the samestability as the fixed pointsof theHénonmapping, independentlyof s. Example3 is aparticularly simple examplefrom classIII of the previous section.Its phaseportrait and Jacobiandeterminantareshown in fig. 4.3a,b (comparefig 4.3awith the phaseportrait of theHénonmappingshown in fig. 3.3). Example 4 is from class IV and is perhapsa less satisfactoryexample of anonconservativereversiblemapping,from the mathematicalpoint of view, becauseit haslines in theplanewhere it andits Jacobiandeterminantarenot defined.Despitethis, the regionsaroundboth itssymmetricandasymmetricfixed pointsdisplaysimilar dynamicalfeaturesto thoseportrayedtogetherinfigs. 4.1—4.3, althoughthe two sets of fixed pointsaremoreseparatedherebecauseof thesmallnessofthe perturbationstrengths (the asymmetricfixed pointsare again“born” from infinity at e = 0).

J.A.G.Robertsand G.R.W. Quispel,Chaos and time-reversalsymmetry 125

32 1

x~0 1

3

2

14 —1- 1

Y x - 0

Re \~ ~

___ 14

(a) (b)

Fig. 4.3. (a) Phaseportrait of themappingexample3 of table 4.1 whenC = 0.5. Thesymmetry line of H is thehumpedcurve shownandthesymmetryline of G is thex axis. Thesymmetricfixed point at theorigin is elliptic andsurroundedby KAM curves.Theattracting(At) andrepelling(Re)asymmetricfixed points aremarkedwith crosses.Externalto the KAM curvesand islandchain areshowntwo trajectoriesthat eventuallyconvergeto theattractingfixed point. (b) Plotof theJacobiandeterminantof example3 over thepartof theplaneshownin a(actuallythedomainhereis [—1.6,1.6] x [—1.6,1.6]).Overthis regionI(x, y) rangesfrom closeto 0 to greaterthan3. Thetoptwo picturesshowthesurfaceobtainedby plottingthe Jacobiandeterminantwhenviewed from two different perspectives.It hasa trough nearthe attractingasymmetricfixed point at(1.2807.. . , —1) andis muchhigheraroundtherepellingasymmetricfixed point at (1.2807.. . , +1). In thebottompicturewe showonly thepartof thesurfacewhereI < 1 (note theexpandedscaleon the I axis).Thesymmetricfixed pointsat (0,0) and(0.5,0), which haveI = 1,arepointsontheprojectionof the boundaryof this partof thesurfaceonto the plane.

126 J.A.G.Robertsand G.R.W. Quispel,Chaosand time-reversalsymmetry

Betweenthem, the four examplesin table 4.1 display a rangeof featuresandphenomena.Theseexampleswill beusedin chapters5 and 6 to illustrate via numericalandpictorial investigationssomeofthe propertiesof general(i.e., non measure-preserving)reversiblemappings.Attention will tend tofocus on examples1—3 because,unlike example4, they are “far” from area-preservingreversiblemappings,and moreoverthey are analyticthroughout theplane without the complicationof havingsingularity lines.

5. Conservativebehaviourin reversiblemappings

In this chapter,phenomenaassociatedwith thesymmetricperiodicorbits of reversiblemappingswillbestudied.We review thework doneon area-preservingreversiblemappingsandpresentresultsforthenon measure-preservingreversiblemappingsof table4.1. As previously remarked,theneighbourhoodof an elliptic symmetricfixed point in the examplesof table 4.1 appearsindistinguishableto the eyefrom that ofan elliptic fixed point in anarea-preservingmapping(cf. fig. 4.2aaroundtheorigin with fig.3.2). What is obviously different betweenarea-preservingmappingsand the mappingsof table 4.1 isthat, althoughthe Jacobiandeterminantat anelliptic symmetricfixed point in the latter equals1, it isnot equalto 1 aroundthefixed point, but variesin a rangeof valueson eithersideof 1 (cf. theJacobiandeterminantplots in the previouschapter).

Herewe investigatewhetherthereareany quantitativeeffectsdueto this lack of areapreservationaroundsymmetricfixed pointsor symmetric cyclesin a generalreversiblemapping.In section5.1 westudy the breakingof the KAM curves,or circles, aroundelliptic symmetricfixed pointsby studyingsymmetricperiodic orbits that lie close to thesecurves. In section5.2 we study the period-doublingcascadethat can ariseafter symmetricperiodicorbits turn from beingelliptic to hyperbolic. Criticalexponentsassociatedwith thesetwo processesinvolving symmetricperiodicorbits are shownto be thesame as those previously associatedwith conservativemappings (cf. also Quispel and Roberts[1988,1989]).

Becausethe phenomenadiscussedhere involve the study of symmetric periodic orbits in one-parameterreversiblemappings,it is importantto realise that anisolatedsymmetricperiodicorbit stayssymmetricas a mappingparameteris varied.The argumentfor this hasbeengivenby MacKay [1982]andrelieson thePoincaréindex (cf. section2.1). If theperiodicorbit haslength n, thentakeL” sothateachperiodic point is a fixed point (of L”). A symmetricfixed point x~hasJ = DetdL(x0)= 1 for Lorientation-preserving,so that its index is + 1 or —1 accordinglyasTr dL(x0)<2 orTr dL(x0) >2 [cf.(2.12a,b)]. If the fixed point were to becomeasymmetricat any stageby moving off thesymmetryline,thenthis eventwould be accompaniedby thebirth of anasymmetricpartneringfixed point. It is easytocheckthat thesetwo asymmetricpoints(which mostgenerallyhaveJ ~ 1) havethesameindexbecausetheir eigenvaluesarereciprocalsof oneanother[cf. (2.12);if (2.12a)is truefor onepoint, it is true fortheotherpoint andsimilarlywith (2.12b)]. The two asymmetricfixed points thereforehavea combinedindexof +2 or —2 which is differentto thetotal indexof + 1 or —1 beforethesymmetricfixed point leftthe symmetryline, contradictingtheconservationof Poincaréindex.

Although numericallytheproblemof finding periodicorbits of an arbitrarymappingof theplaneLin generalrequiresatwo-dimensional(2D) searchfor solutionsof eqs.(2.5), thesearchfor symmetricperiodicorbits in a reversiblemappingL = Ho G is a 1D search.Welook alonga curvein the plane,namelyasymmetry line, andour trial pointsfor the searchareparametrisedby one coordinate.If, forexample,thesymmetryline hastheequationy = h(x) thenonly one independentvariablex appearsin

J.A.G. Robertsand G.R.W. Quispel,Chaosand time.reversalsymmetry 127

the trial point (x, h(x)). A point of asymmetricn-cycleon thesymmetryline can becalculatedby, forexample,repeatedapplicationsof thesecantmethod.By fully exploiting thebenefitsof reversibility(cf.section3.1), symmetricn-cyclescan be locatedwithout the need to traversethe entire cycle. Evenn-cyclescan be found by going halfway aroundthe orbit and using the condition that y,,

12 = h(x~,2),i.e., thehalfwayroundpoint is necessarilyon thesamesymmetryline asthe initial point. Oddn-cyclescan be more economicallyfound by noting, for instance,that (x(~+ 1)/2’ Y(n+ 1)/2) lies on the symmetryline of H if the initial point is on thesymmetryline of G. It is a furtherconsequenceof thepropertiesofsection3.1 thatalthough a reversiblemappingcanbewrittenin manydifferent waysasthecompositionof involutions by using different symmetriesfrom a family [e.g.,L = (L’ + 10 G) 0 (GoL -i )], we canexpect to find any symmetricperiodic orbit by searchingupon the two symmetry lines of the twoinvolutions that provide a nice decompositionfor L, like thoselisted in table4.1.

5.1. KAM circlesand their destruction

The outline of this sectionis as follows. We first review the KAM theoremsfor area-preservingmappingsand for reversible mappings(section 5.1.1). This leadsto a discussionof the (irrational)rotationnumberof invariantKAM circles and its approximationby a sequenceof rationalconvergents.We thendescribea quantitativetechniquethat usessymmetricperiodic orbits with rotationnumbersequal to theserational convergentsto study a given KAM circle (section 5.1.2). This technique,Greene’sresiduecriterion (cf. Greene [1979],MacKay [19921),was originally developedfor area-preservingreversible mappingsbut we discussthe application of this techniqueto non measure-preservingreversiblemappingsandpresentsomenumericalresults.

5.1.1. Conservativeand reversibleKAM theoremsThe most orderedof dynamicalsystemsare the integrablesystemsbecausethey do not haveany

chaoticmotion. Their dynamicstypically consistsof periodicand quasiperiodicmotion. The canonicalintegrablemappingof the planeis the integrabletwist mapping

r’r, O’=O+w(r). (5.1)

This mappingin polarcoordinatesleavesinvariantall circles centredon theorigin so that the radiusr isan integral or “constantof the motion”. The quantity p(r) = w(r) /2ir E (0, 1), is called the rotationnumberof a given circle and most generallyvaries from circle to circle. It completelydescribesthedynamicson the circle of radiusr = r0. Whenp(r0) = p/q for p, q relatively prime,theorbit of everypoint on the circle is periodicof length q andinvolvesp completerevolutionsaroundtheorigin. On theotherhand,if p(r0) is irrational, theorbit of eachpointdenselyifils the circle and is quasiperiodic.Theintegrabletwist mappinghasthedistinguishingpropertythat it hassetsof closedinvariantcurveswhoserotationnumberscompletelyfill someinterval. Thereareno gapsatthe rational rotationnumbersandthesecorrespondto invariant curvesof non-isolatedperiodicpoints.

The mapping (5.1) is area-preserving.It is also reversiblewith respectto the involution r’ = r,0’ = —0. Indeed,(5.1) is simplythepolarform of theformal normalform (3.59) aroundanelliptic fixedpoint of a measure-preservingmapping. Integrablemappingslike (5.1) are special and rare— seeappendixA for someotherexamplesof integrablereversiblemappings.Their significanceis nonethe-less assuredby the KAM theoremsthat statethat systemsclose to integrableretain some of theimportantpropertiesof integrablemappings.

128 I.A.G.Robertsand G.R.W.Qutspel,Chaosand time-reversalsymmetry

The starting point of the KAM theoremsfor mappingsof the planearisesfrom a considerationofwhat happensto the dynamicswhen (5.1) is slightly perturbedby the addition of small functionsofperiod2 ir in 0 to give the mapping

= r + a(r, 0), 0’ = U + w(r) + b(r, 0). (5.2)

The result when the perturbing functions in (5.2) are such that the perturbedmapping is stillarea-preserving,havebeenknownfor a long time [Moser1962, 1968; Siegel andMoser 1971]. In thiscasethe invariantcirclesof (5.1) with rationalrotationnumberp = p/q typically breakup to leavejusttwo cycleswith thatrotationnumber(typically a mappingof theplaneis not integrableandso doesnothavean infinity of cyclesof a certain length, cf. Moser [1968]).Theseare the so-calledPoincaré—Birkhoff cyclesand they haveoppositestability. Someof the invariant circlesof (5.1) with irrationalrotationnumber remain for theperturbedmapping,though they are deformedinto closedinvariantcurves homeomorphicto circles. The motion on thesedeformedcircles is still quasiperiodicwith thesamerotationnumberasin theunperturbedmapping.In fact, the set of deformedinvariantcircleshasfinite measureandcoversan increasingfraction of the areaaroundthe fixed point at theorigin if oneconsiderssmallerand smallerneighbourhoodsaroundthis point.

Other invariantcircleswith irrational rotation numberdo breakup on perturbation;and the largertheperturbation,thesmallerthemeasureofinvariantcurvesthat remain.Thecriterion thatdetermineswhetheracircle breaksor not underperturbation,andwhenit breaks(as a function of thesizeof theperturbation),is the “degreeof irrationality” of its rotationnumber.This will be discussedbelow.

Originally the above result, which we will call the global KAM theorem for area-preservingmappings,hadthe mathematicalrequirements:(i) that the functionsa(r, 0) andb(r, 0) be real andanalytic;and (ii) that w(r) beanalyticandmonotonicincreasingso that theunperturbedmapping(5.1)bent a given radius vector uniformly in the one direction (it was a monotonetwist mapping).Thedomainof themapping (5.2) could eitherbe adisc centredon theorigin or the annulusc � r � d (notnecessarilymappedinto itself). Laterwork hasrelaxedthedifferentiability conditionson the functionsa andb so that the resultstill holds (i.e., invariantcurvesstill persist) if thesefunctionsareonly C3,which is in facttheminimum possibledifferentiability requirementon themfor which the theoremstillholds [Herman1983].

Significantly, the global KAM theoremhaswide applicability to very many area-preservingmap-pings,notnecessarilythosethat areslight perturbationsof the integrablemapping(5.1).This is becausein the neighbourhoodof a “typical” elliptic fixed point of an area-preservingmapping,the mappingtakesthe form (5.2). Here “typical” means,for a mappingthat is at leastC4 differentiable,that theeigenvaluesof the fixed point arenot first, second,third or fourth roots of unity [Siegeland Moser1971; Moser 1968, 1973]. This applicationof theglobal result,which we call the local KAM theorem,implies that an elliptic fixed point of an area-preservingmappingis surroundedby invariant curvesaswell as island chains associatedwith elliptic Poincaré—Birkhoffcycles. Interspersedbetweenthesecurves are chaotic bandsassociatedwith the transversehomoclinic intersectionsof the stableandunstablemanifoldsof hyperbolic Poincaré—Birkhoffcycles.

The recentKAM theoremsof Arnol’d andSevryukfor reversiblemappingsalso exist in both globaland local versions[Arnol’d 1984; Arnol’d and Sevryuk 1986; Sevryuk 1986]. The global theoremconsiderstheperturbedtwist mapping(5.2) whena(r, 0) andb(r, 0) arechosenso that theperturbedmappingremainsreversiblethoughnot necessarilyarea-preserving.It requirestheperturbingfunctions

lAG. Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry 129

to be analytic.Theconclusionis muchthesameasin thearea-preservingcase*);a largeproportion(afinite measure)of the invariant curves of an integrable reversible mapping persist under a smallreversibleperturbation.This resultis illustratedin fig. 5.1 with anintegrablereversiblemappingtakenfrom Quispelet al. [1989].Accordingto Arnol’d andSevryuk,the rationalcirclesof (5.1) breakup tostill leave at least two cycles with each rational rotation number. Thesecycles are symmetric (cf.Sevryuk [1986,chapter 5]) and they constitute the Poincaré—Birkhoff cycles for the (reversible)perturbedmapping.*

The local KAM theoremfor reversiblemappings,with which wewill bemoreconcerned,guaranteestheexistenceof infinitely manynestedclosedinvariantcurvesaroundanonresonantelliptic symmetricfixed point. MoreoverthesecurvesareinvariantunderthesymmetryG aswell asunderthemappingL.Again the local result follows from the global one by noting that closeto an elliptic symmetric fixedpoint the motion is locally conjugateto a perturbationof integrablemotion.

The determiningfactor in the (reversibleor conservative)KAM theoremsto establishwhether aparticularinvariantcurveof (5.1) persistsunderperturbationis its rotationnumber.The “robustness”of a givencurveis directly relatedto howirrational its rotationnumberis. The circlesof (5.1) that dopersistundersufficiently small (reversibleor conservative)perturbations(5.2) are those circles with“sufficiently irrational” rotationnumberp satisfyingrelationsof the form

I~—p/q~ � K/(q2~), (5.3)

for all integersp and q >0 and for somepositive constantsK and r.A large fraction of invariantcircles remain becauseof thenumber-theoreticpropertiesof irrational

numbers.The condition (5.3) is an exampleof a Diophantinecondition and is satisfied by mostirrational numbers(for small K). It is a statementof how well the irrational number p can beapproximatedby nearbyrationals.It is knownthat any irrationalnumberp canbe bestapproximatedby the sequenceof rationalsp~= p,,/q~,calledtheconvergentsof p, which areobtainedasthenth leveltruncationsof its continuedfraction expansion(CFE),

1 1 =: [n1,n2, n3,. . .], (5.4)

1

+

n3 +

where n. are positive integers. Every irrational p E (0, 1) has a unique representationas such acontinuedfraction— see Niven [1956]for a discussionand the methodfor generatingthe CFE. Theirrationalnumberlies betweenany two successivetruncations,so that the convergentsapproximateit

*) TheKAM theoremprovedby Moser[1962]is typically appliedto area.preservingmappingsbut in fact thetheoremholds if theperturbedmapping(5.2) is presumedinsteadto havethe self-intersectionproperty,which saysthat it mapsany closedcurvein theannulusdomainin suchaway that thecurve intersectsits image (seeSiegel and Moser [1971,sections32—34]). Although in generalreversibility doesnot imply thisintersectionproperty, Sevryuk[1986,p. 57] statesthathedoesnot know of any reversiblemappingarbitrarilycloseto thetwist mapping(5.1)thatdoesnot possesstheproperty (seealsoSevryuk [1986,p. 154]).

**) Veryfew theoremsexist concerningthenumberandarrangementof symmetriccyclesalongsymmetrylinesin non area-preservingreversiblemappings.Oneexception,from TanikawaandYamaguchi[1987],saysthatbetweenthetwo pointsof asymmetric21-cycleon asymmetryline liesaperiodic point of periodj.

130 J.A.G. Robertsand G.R.W.Quispel, Chaosand time-reversalsymmetry

(a)

40

- : ~ ~

(b)

Fig. 5.1. (a) Phaseportrait of the mappingx’=y, y’=[2Ay—x (1—A2y2)]/(1—A2y2+2Axy)with A=0.25. This mapping is an integrabte

reversiblemapping(cf. Quispeletxi. [1989]).The“bow-tie” shapedclosedcurvesareall invariantunderthemapping,asaresetsof two pairsof theunclosedcurvesthat are symmetricabouty = x. (b) Phaseportrait of themappingx’= y, y’= [2Ay+ py3 — x (1— A2y2)]I(1 — A2y2+2A.~i),forp= 0.0005.This mappingis’a nonintegrablereversibleperturbationof the integrablemappingshownin a. It still appearsto have invariantcurves.

J.A.G. Robertsand G.R.W.Quispel, Chaosand time-reversalsymmetry 131

from either side. Furthermoresuccessiveconvergentsp~/q~andp,~+ 1/q~+ 1 areneighbouringrationals,

i.e.,

p~q~~1—p~÷1q~= ±1. (5.5)

The “most irrational” irrationals, those that are poorly approximated(relatively speaking)by theirrationalconvergents,arecalledthe “noble numbers”(cf. Series[1985]for a readablediscussion).Thenoble numbershaveCFE’s ending in 1, 1, 1 The canonicalexampleof a noble numberin theinterval (0, 1) is thenumber

(5.6)

which is the reciprocalof thegoldenmean,

(5.7)

Thenumber(5.6) hasa CFE (5.4) that consistsentirelyof l’s. Onefinds by successivelytruncatingitsCFE that its rationalconvergentsaregiven by F~/F~+1 whereF,, aretheFibonaccinumbers:1, 1, 2, 3,5, 8, 13, 21,... given by F,,~2 = F,,~1 + F,,.

In the interval (p/q, p’/q’), where p/q and p’/q’ are any neighbouringrationalswith q’ > q, thenumber

(p + yp’)/(q + -yq’) (5.8)

is noble. Its rational approximantsare

p01q0=p/q, p1/q1=p’/q’, p2/q2—(p0+p1)/(q0+q1), (5.9)

andin general

p,,+1/q,,÷1=(p,,_1+p,,)/(q,,_1+q,,). (5.10)

Thus the approximantsare generatedby a Farey tree construction.

5.1.2. Breakupof KAM circles; the residuecriterionInvariantcurveswith noblerotationnumber,beingthecurveson which the rotationnumberis most

strongly irrational, certainlysurvivethe small perturbationsfrom integrabilityallowedin the conserva-tive andreversibleKAM theorems.In additionhowever,they figure prominentlyin numericalstudiesof KAM curvebreakupwhenthe size of theperturbationexceedsthe realmof the analyticaltheory.This is becauseGreene[1979]found numerically, for reversiblearea-preservingmappings,that oftenthe invariantcurveswith noblerotation numbersare the most robust; if an invariantcurve of somenon-noblefrequencyexists, thenthereis also anobleinvariantcurve (cf. SchmidtandBialek [1982]).An approximaterenormalisationschemealso leadsto this conclusion[Escande1982]. It hasbeenconjectured(cf. MacKay [1986])and recently proved undersomeconditions (cf. MacKay andStark[1991])that for typical one-parameterarea-preservingmappings,the last invariantcurveto breakin an

132 iA. G. Robertsand G.R.W. Quispel, Chaosand time-reversalsymmetry

intervalof rotationnumber (p/q,p’/q’), wherep/q andp’/q’ areneighbouringrationalswith q’ > qand the interval is short, is the curve with rotation number (5.8). The interval of rotation number(p/q,p ‘/q’) canbe translatedinto anappropriateregionof phasespaceby locatingtwo periodicorbitswith rotationnumbersp/q and,p ‘/q’, respectively.

The numericalmethoddevelopedby Greeneconjecturesa relationshipbetweenthe linear stabilityof symmetricperiodicorbits nearbya KAM curveof an area-preservingmapping and the existenceornon-existenceof the invariantcurve. The periodicorbits usedarepreciselythose(Poincaré—Birkhoff)cycleswith rotation numbersp,, = p,,/q,,equalto theconvergentsof theirrational rotationnumberp ofthe curve. We will variously call these particular periodic orbits the (noble) approximating, orapproximant, (periodic) orbits or cycles.The methodis calledthe residue criterion, wherethe residue,R, of a periodic orbit with period q,, is given by

R= ~(2—Tr[dL”]). (5.11)

When0<R< 1 theorbit is elliptic, andwhenR <0orR> 1 theorbit is hyperbolicor morespecificallya saddle,so it is unstable.The transitioncasesR = 0 and R = 1 correspondrespectivelyto a pair ofeigenvaluesat + 1 anda pair of eigenvaluesat —1. The two Poincaré—Birkhoffcyclesfor eachrationalrotationnumberaresuchthat one haspositive residueand theother negativeresidue.

Denoteby R,, thepositive residuesof thesymmetriccycleswith rotationnumbersp,, = p,,/q,,—~ p atagiven value of a mapping parameterC. Then Greene’s conjecturebased upon observationsforarea-preservingreversiblemappingsis that asn —~ oneof the following occurs:

R~—~ 0~and thereis a smoothinvariantcurve with rotation numberp at C; (5. 12a)

R,, —p ~ and thereis no curve with rotation numberp at C; (5.l2b)

R~—~ RKAM, 0<RKAM < 1 at C = CKAM and thereis a non-smoothinvariantcurvewith rotation

numberp. (5.12c)

In case(5.12b) the nearbysymmetricperiodicorbits all haveR,, > 1 beyondsomen, or equivalentlybeyondsomeorbit length q,,, and so are unstable.Case(5.12c) is thecritical (transition)case,in thesensethat (5.12a) occurson oneside of CKAM and (5.l2b) occurson theother.The interpretationgivento this behaviourof the residues,evidencefor which includescarefully assembledpictorial investiga-tions [Greene1979; Shenkerand Kadanoff1982], is that in case(5.12a) thesuccessiveapproximatingperiodiccyclesappearto lie on smoothcurvesthat convergefrom eitherside to a smoothKAM curve.In case(5.12b),theapproximatingorbits areon jaggedcurvesthat seemto precludetheexistenceof aKAM curvebetweenthem— theKAM curvehasbrokenup by developinginfinitely manygapsin it andhas self-similar structure like a Cantor set (dubbed a cantorus, cf. Percival [1979])*).At theintermediatecritical stage (5.12c) when C = CKAM, the approximatingorbits of longer and longerperiodsappearto lie upon curveswith self-similar featuresupon magnification. [Notethat CKAM canalso be identifiedby studyingthenegative-residuePoincaré—Birkhoffcycleswith rotationnumbersp,,;their residuesR~behaveanalogouslyto that listed in (5.12a—c),with, respectively,0+ —* 0, +co—+ —re

~ We remark that recent work of Hu et al. [1991]showsthat in nonanalytic(C’) twist mapsaKAM curvecanreappearafter it hasbrokenup;theyalso report different residuebehaviourin somecasesfrom that listedin (5.12).

JAG. Robertsand G.R.W.Quispel, Chaos and time-reversalsymmetry 133

and RKAM, 0<RKAM < l—~R~AM,—1<R~AM<0]. Very recently,some aspectsof the relationshipbetweenthe residuesand the existenceof the KAM curve havebeen maderigorous (cf. MacKay[1992]).

The existenceof a local KAM theoremfor reversiblemappingsas describedin section5.1.1 abovemakes it natural to study the residuecriterion in reversible(non measure-preserving)mappings(cf.Quispeland Roberts[1988,1989]).We havestudiedthedestructionof an invariantnoblecurvearoundtheelliptic symmetricfixed point at theorigin in example3 of table4.1, andat (1 — C, 0) in example4.The chosennoble rotationnumberis the reciprocalof thesquareof the goldenmean,

PKAM72(3~)0.3819660h1~, (5.13)

which hastheCFE [2,1, 1, 1,. . .]. This numberis of the form (5.8) with plq = 1/2 andp’Iq’ = 1/3. Itsrational approximantsfrom (5.9) and (5.10) turn out to be ratios of secondsuccessiveFibonaccinumbers,i.e.,

p~=p,,/q,, = F,,/F~±2. (5.14)

As in studiesof area-preservingreversible mappings,we locate the positive-residuesymmetricPoincaré—Birkhoff periodic orbits with rotation numbers (5.14) and “plot” their residuesas theparameterC is varied.Of course,without areapreservation,the returnJacobiandeterminantof theseorbits still equals1 becauseof their symmetryand (5.11) still completelydescribestheir stability. Thebestway to locate theseorbits is to startsearchingfor the shortestapproximantorbits [e.g., 1/3, 2/5,3/8,5/13 for PKAM = (3 — \/5)12] on the symmetrylinesof H andG. The positionandrotationnumberof thesesmallerperiodapproximantorbits canbe usedto give goodinitial guessesfor the positionsofsuccessivelylonger approximantorbits. This is becausethe rotation number is found to be locallymonotonicasa function ofpositionalongthesymmetrylines so that at eachvalueof theparametertheposition of an approximantperiodic orbit can be estimatedby linearly interpolating betweenthepositionsfound for the previoustwo approximants(notethat the rotationnumberof eachapproximantorbit lies betweenthoseof the two previ6usapproximants).

In area-preservingmappingsit hasbeenobservednumericallythat it is alwayspossibleto considerthe symmetry lines of G andH which intersectat the symmetricfixed point as four half-lines, and tofind onepointof everyapproximatingpositive-residueorbit on just oneof thesehalf-lines(cf. Greene[1979],MacKay[1986],Meiss [1986]);sometimesthisis only possiblefor the orbits beyondsomelength(cf. KetojaandMacKay [1989]).That this should alwaysbe possible is called the “dominantreversorconjecture” for the residuecriterion. We find that it appearsto hold in non measure-preserving,reversiblemappingsas well.

Once the approximantperiodic orbits are locatedfor one value of the mappingparameterC, theparametercan be incrementedand the motion of the orbits on the symmetry line can be followed.Calculationof the residueof theorbit for eachparametervalueis doneby multiplying the Jacobianmatricesevaluatedat eachpoint of theorbit to give the returnJacobianmatrix and thenby using thedefinition (5.11). In suchaway, the residuecurveasa function of parameterfor the particularperiodicorbit is constructed,cf. fig. 5.2. The qualitativebehaviourof the residuesin fig. 5.2 is consistentwithGreene’sobservationsdescribedin (5.12a)—(5.12c) above.Quantitativecomparisonswith the resultsforarea-preservingmappingscan also be made. A numericalstudy of the breakup of noble curvesinseveral one-parameterfamilies of area-preservingreversiblemappingshasrevealedsome universal

134 J.A.G. Robertsand G.R.W. Quispel, Chaosand time-reversalsymmetry

1.0001597 2584

987

0.750 610

.377= 0.500

0.250~

0.000• I

-0.89000 -0.88980 -0.88960 -0.88940 -0.88920 -0.88900

ParameterCFig. 5.2. Residuecurves for symmetriccyclessurroundingtheorigin in example3 of table4.1 with periods377, 610, 987, 1597 and2584. Therotationnumbersof thesecyclesare respectively144/377,233/610,377/987,610/1597and987/2584.This sequenceof rational numbersconvergesfrom eitherside to the noble irrational number(3— V3)/2.

quantitativebehaviourandasymptoticscaling[MacKay1986]. Wefind identicalbehaviourfor the nonmeasure-preservingreversiblemappingsof chapter4, in particular:

(i) The valueof RKAM in (5.12c) is the samefor eachof thesemappingswith value

RKAM =0.25008... (5.15)

This valuecanbe obtainedasthe limit of theresiduesat thepointsof intersectionofthe residuecurvesR,,(C) and R,,~

1(C) of successivepairs of rational convergents. Denote the sequence of theseintersection residues by ~ Then the convergenceof R~~tis asymptoticallygeometric,i.e.,*)

R~1— R”

öR ~ (5.16)

The negativevalue of ~RKAM indicatesthat R~1 lies betweenR~i1andR~t.This is illustrated in fig.

5.3.

(ii) The valueof CKAM can be found from the sequenceof parametervaluesC,~’at thepointsofintersectionof the residuecurvesR,,(C) and R,,+ 1(C). This sequenceconvergesto a value CKAM

asymptoticallygeometricallywith a universal ratio so that

*) We adopt the common convention of saying that somethingconvergesgeometrically but quoting the inverseof the geometric ratio. This

inverseis a numbergreaterthan one in absolutevalue.

JAG. Robertsand G.R.W. Quispel, Chaosand time-reversalsymmetry 135

0.2575

0.2550

0.2525

1597 377

0.25000.2475 987

0.2450 2584

610

0.2425-0.889340 -0.889335 -0.889330 -0.889325 -0.889320

ParameterCFig. 5.3. Enlargementof fig. 5.2aroundtheareain which theresiduecurvesappearto all intersect..In factthis is not thecase,butwe can look atintersectionsbetweenthecurvesof successiveapproximantcycles.Herewe indicateby arrowstheintersectionof the residuecurvesofperiods610and987 (right-handarrow), 987 and1597 (left-handarrow)and1597 and2584 (middlearrow). This illustratesthepatternthat theparametervalueandresiduevalueat anintersectionof the residuecurvesof two successiveapproximantsarecontainedbetweenthosevaluesat the previoustwointersections.

Ci~~t— C1n1

= C~1— ~ = —2.665... (5.17)

Note that thesescalingscould havebeenobtainedby alternativemethodsas detailedby MacKay[1986].Someresiduebehaviourdifferent to that describedabovewill be discussedat the endof thissection.

To specifically illustrate the determination of the quantities (5.15).-(5.17),table 5.1 presentsnumericalresultsfor thebreakup of thecurvewith rotationnumber~2 given by (5.13)in example3of table 4.1. Listed are the sequencesR~’

tandC~tfor successiverationalapproximantperiodicorbits(5.14) [note:rotation numberfor theseorbits = (revolutionsaroundthe origin)/period],as well astheposition of one of the points of each approximantorbit on a symmetry line (the numberof figuresquoted for the coordinatereflects confidencein agreementwith the true value).For eachtriple ofconsecutiveC,” valueswe calculatethe valueö~definedby (5.17). The number ~5R on each line iscalculatedsimilarly, substitutingthe R’,,” valuefrom the line togetherwith the valuesfrom the linesimmediatelyaboveand below into (5.16).The sequencesof valuest5~ andö~indicatethat thevaluesof R~~tand C~~tappearto convergeasymptoticallygeometrically,respectively,from aboveandbelowtoa value RKAM, and from left and right to a value CKAM. We estimatethese limiting values bysuperconvergingeachsequence(using Aitken’s method, cf. Abramowitzand Stegun [1972,section3.9.7]).

Theselimiting valuesarelisted in table5.2, togetherwith thevaluesof 8CKAM and öRKAMobtainedfrom the sequencesô~and 3R’ againby superconverging.Also shown arethe figures obtainedfrom


Table 5.1Intersectionsof residuecurvesof periodicorbits aroundthe symmetricfixed point (0,0) of example3 oftable4.1. TheparametervalueC~’”should

give agreementbetweenthe residuesof eachpair of orbits shownto at leastthenumberof figures in R,”.

Period Revolution Coordinate’~ Sym~ C~” R’,~”

377 144 0.104587420634... 1

610 233 0.104586361402... 1 —0.8893625536660790 0.2620108...

610 233 0.104577979034... 1987 377 0.104578328862... 1 —0.8893254348580585 —2.6659578. .. 0.243081... — 1.662...

987 377 0.10458147116... 11597 610 0.10458135798... 1 —0.8893393581118690 —2.6652111... 0.254471. .. —1.622...

1597 610 0.10458017858... 12584 987 0.10458021566... 1 —0.8893341340403813—2.6761848... 0.247449... —1.655...

2584 987 0.10458065628... 14181 1597 0.1045806442... 1 —0.8893360860996619 —2.7881624... 0.25169... —1.703...

4181 1597 0.10458048618... 16765 2584 0.10458049012... 1 —0.8893353859757207 0.2492...

‘~“Sym” = 1 correspondsto y3— y2 +y+ Cx +x2 — x= 0 and “coordinate”is thex coordinateon this line.

Table5.2Numerical results for residue scalingsassociatedwith noble KAM curvesin reversible

mappings.

Non measure-preservingreversiblemappings~

Area-preservingExample3 Example4 reversiblemappings”~

RKAM 0.2501 ... 0.25008... 0.2500888...ÔRKAM —1.6.. . —1.637... —1.6371161...SCKAM —2.6.. . —2.665... —2.6651429...CKAM —0.88933555... 2.841404026...

~ Theseresultsareobtainedfrom double-precisioncalculations.~,) Theseresultsare takenfrom MacKay [1986]andaretheresultof quadruple-precision

calculations.

similar calculationsfor example4 oftable4.1,aswell astheresultsfoundfor area-preservingmappings.From table 5.2 we concludethat the valueRKAM and the scalings3R,KAM and ÔC,KAM associatedwithnoble KAM curves in non measure-preserving,reversiblemappingsareuniversal,and moreoverthesameasthoseuniversalvaluespreviouslyfoundfor area-preserving,reversiblemappings.ThevaluesofCKAM are of coursepeculiarto themapping and notuniversal.

We conjecturethat thebehaviourobservedcarriesover to relatingthebehaviourof the residuestothe existenceof the noble curve as done by Greene[1979].This is basedupon the qualitativesimilaritiesshown in fig. 5.2 and the quantitativeagreementshownin table5.2, and on the fact thatGreene’sconjecturewould seemto rely on the topologicalpropertyof howwell theapproximantorbitsfit the KAM curve, which would not appearto be affected by the local Jacobiandeterminantenvironment.


Finally we come to the theoreticalexplanationof the universalbehaviourassociatedwith nobleKAM curve destruction,at least for area-preservingmappings, for which such theory has beendeveloped.The theoreticalexplanationis in termsof renormalisationgroup theory. This topic forarea-preservingmappingshas alreadybeenthoroughly covered— seeShenkerand Kadanoff [1982],MacKay [1982,1983b,1986, 1988], Escande[1985],Greene [1986]and referencestherein— so wementionit only briefly.

In the renormalisationpicture, the existenceof KAM curves with noble rotation number inarea-preservingreversiblemappingsis explainedby thepresenceof a simple attractingfixed point of acertainrenormalisationoperatorin thespaceof (pairsof) area-preservingmappings.The simple fixedpoint mappingis integrablelike (5.1) andpossessesaninvariantcircle with noblerotationnumber.Thequestionof whethera given (nonintegrable)mappingpossessesa noble circle amountsto whetherthismappinglies in the basinof attractionof the simple fixed point. Mappingsattractedto this fixed pointhavenoble approximantperiodic orbits with residuesR,, —*0. Part of the boundaryof the basin ofattraction of the simple fixed point comprisesthe stable manifold of anotherfixed point of therenormalisationoperator,the so-calledcritical fixed point, which hasa critical noblecircle. Mappingsattractedto this fixed point along its stablemanifold havenoble approximantperiodic orbits withresiduesR,, —* RKAM = 0.250088. . . The critical residuescalings~C,KAM and 5RKAM given in table 5.2abovearerelatedto theeigenvaluesof the linearisedrenormalisationoperatoraroundthis critical fixedpoint.

Recentwork hasconcentratedon further defining the basin boundaryof the simple fixed pointmapping. This has led to the discoveryof other area-preservingmappingsthat are fixed points orperiodic orbits of the renormalisationoperator[Greeneet al. 1987; Johannessonet a!. 1988; KetojaandMacKay 1989; GreeneandMao 1990; Greene1990;Wilbrink 1990]. Mappingson thestablemanifoldsof suchotherfixed pointsor cycleshavedifferentresiduebehaviourto that listed above.For instance,the sequenceof residuesR,, of noble approximantperiodic orbits may convergeto a three-cycleofvalues,none equalto 0.250088...,correspondingto a three-cycleof the renormalisationoperator[Greeneand Mao 1990; Wilbrink 1990]. Such behaviourappearswhen the mappinghas additionalsymmetry, or equivalently,commuteswith a nontrivial mapping [MacKay1984], for exampleif themappingis odd so that L(—x) = — L(x) (e.g., Huiszoon[1983]).Whenthe mapping is reversible, theexistenceof nontrivial commutingmaps often leadsto it being multiply reversible (cf. 3.7a,b) withconsequentlymorereversingsymmetriesand symmetry lines (cf. Wilbrink 11990]).

It would be desirableto extendthe renormalisationgrouptreatmentto encompasstheaboveresultsfor reversiblemappingsthat are not area-preserving(cf. also Khanin andSinai [1986]).

5.2. Perioddoublingof symmetricfixedpoints

The pointsof transitionbetweena symmetricfixed point beinghyperbolicorelliptic occurwhenitseigenvaluesareboth + 1 orboth —1. Thesetwo casescorrespondto the traceofthe (Jacobianmatrix atthe) fixed point equalling+2 or —2. In thepresentsectionwe discussthecasewherebotheigenvaluesof a symmetricfixed point of a one-parameterreversiblemapping (symmetricwith respectto G, say)passthrough —1 (i.e., its traceis —2) and the pointturnshyperbolic.For everyexamplein table4.1 itis observedthat a symmetric two-cycle is born at this stage (cf. fig. 5.4). The eigenvaluesof thetwo-cycle at creation are both + 1 as requiredby continuity (the eigenvaluesof the fixed point at


026

Y

—0 26 ~ 3 26

Q~26

Fig. 5.4. Regionaroundthe origin in example2 of table 4.1 at C = —1.02, soonafter thesymmetricfixed point therehasturnedhyperbolicandgivenbirth to asymmetrictwo-cycle lying on thesymmetry line of H. Thetwo-cycleis elliptic andsois surroundedby itsown invariantcurves(thenesteddoubleislands).Someof theouter invariantcurvesthatenclosedtheorigin whenit waselliptic appearto still existeventhoughthepointisnow hyperbolic.

bifurcationare +1 when it is consideredas a fixed point of L2 ratherthanof L). *) The two-cycle isfoundto haveboth of its pointson the symmetryline of G or on the symmetryline of H = L ° G.

With thecontinuousvariation of the mappingparameterC wefind that, after it is born (with traceequalto 2), the eigenvalues of the two-cycle become complexsothat the two-cycle is elliptic. They thentypically traversethe unit circle in the complexplane,passingthroughthe so-calledsuperstablepoint(traceequalto 0) wherethey are±i,andcoincidingagainwhen theyareboth —1 (traceequalto —2).At this time, the two-cycle turnshyperbolicand gives birth to a symmetricfour-cycle. The four-cyclehas two points on the samesymmetry line as the two-cycle. These two points are-born alongthissymmetry line from one point of the two-cycle; theotherpoint of the two-cycle bifurcatesacrossthissymmetryline to producethe othertwo pointsof the four-cycle.

We wish to study herethe numerical observationthat this processcan apparentlyin somecasescontinuead infinitum. That is, by continuouslyvarying themapparameter,successivesymmetricperioddoublings continue with a symmetric 2~-cyclegiving birth to a symmetric 2~~1-cyclewhen itseigenvaluesare both —1. In this way, a cascadeof evensymmetriccyclesis formed.The most recentcycle is elliptic with —2<trace<2 andso is surrounded by KAM curves;the previouscycles in thecascadeare hyperbolicwith both eigenvaluesbeingnegative(cf. fig. 5.5).

Thesequalitativeobservationsareexplainedby theanalysisof MacKay [1982]who hasshown thatasymmetric fixed point in a reversible(area-preservingor non area-preserving)mapping L genericallyperiod-doublesto give a symmetrictwo-cycle on thesymmetryline of G orof L o G. Eachpointof thesymmetrictwo-cycle is itself a symmetricfixed point of the reversiblemapping 2 Consequentlyif the

*) Fromour discussionbelow(2.11)of section2.1,we know that theonly time when atwo-cyclecouldbe bornfrom afixed pointwith J = I iswhenthe traceof thefixed point equals—2.


046

—O4C 0~46

-046

Fig. 5.5. Phaseportraitof example2 of table4.1at C = —1.27294... ,furtheralongthesymmetricperioddoublingcascadearisingfrom theorigin.Now thereexistsa two-cycle (A), afour-cycle(+)andaneight-cycle(S). Theeight-cycleis elliptic but its invariantcurvesaretoosmall to see.It isjust aboutto turn hyperbolicandbifurcateto a16-cycle.Thetwo-cycleandfour.cyclearealreadyhyperbolicbut still remainsymmetric.Theperioddoubling occursalongandacrossthe symmetryline of H with each cyclehaving two pointson this line. Note thex—y symmetry of eachcycle.

two-cycle thenperiod-doubles,one point of it must bifurcate.alongthe symmetry line of G and theotherpoint mustbifurcatealongthesymmetryline of L2 o G, becausewe know that a symmetriccyclecan haveat most two points on the one symmetry line and doesso only when even. The continuedapplicationof this resultgives the symmetricperiod doublingcascadethat is observed.

After MacKay, the symmetryline on which eachorbit of theperioddoublingcascadehastwo pointsis called the dominant symmetryline. The point of eachcycle that period-doublesalong the line iscalled the “good” point. The point that bifurcates acrossthe line is called the “bad” point and ishalfway aroundthe periodic orbit from the good point. The dominantsymmetry line is not uniquebecauseif themost recentcycle ofthecascadehasperiod2’, and it and thepreviouscyclesall havetwopointson the symmetryline of G, say,thenall cycleshavethesametwopointson thesymmetryline ofL”o G wherek is any multiple of 2’.

For many one-parameterfamilies of area-preserving,reversible mappingsthe symmetric perioddoubling process has been quantitatively followed along a dominant symmetry line and universalself-similarbehaviourhasbeenobserved[Bountis1981; Greeneet al. 1981; MacKay 1982, 1983a].Ourinvestigationof symmetricperioddoubling in non measure-preserving,reversiblemappingsrevealsthesamescalingbehaviour.Our resultscan be summarisedas follows:

(i) Define C,, asthe parametervalue at which the 2~-cyclebecomesunstableandbifurcates (in aslight abuseof notationwe will usethe subscriptn to mean2~,i.e., theperiodof the cycle). Thenthesequence of successiveparametervaluesC,, convergesto a value CPDSYM.

(ii) The convergenceof theparametervaluesC,, is asymptoticallygeometric,i.e.,

~ (5.18)

140 JAG. Robertsand G.R.W. Quispel, Chaosand time-reversalsymmetry

where öPDSYM has a universalvalue independentof the mapping. Actually, (5.18) is equally wellobtainedby using the C,, valuesat which eachcycle in thecascadeis superstable(hastrace= 0). Wedifferentiatebetweenthe two sequencesC,, of parametervaluesby using C,,blf andC~UP.

(iii) The successivedistancesdan betweenthegood andbad pointsof eachcycle on the dominantsymmetry line measuredat C,, also form asymptoticallya geometricprogression,i.e.,

a,, = da,,/da,,+i_* aPDSYM 4.018..., (5.19)

where aPDSYM is anotheruniversalconstant.It is observedthat betweenconsecutivecyclesthe goodpoint alternatesbeingtheleft-mostor right-mostpoint of the two pointsof eachcycle on thedominantsymmetry line (cf. fig. 5.6 where this is shownin example2).

(iv) The distancesbetweenthepointsone-quarterand three-quartersthe way aroundeach2~-cycle(n � 2) from thegoodpoint form anasymptoticallygeometricprogressionwith universalratio f

3pD,SYM’

i.e.,

f3,, = dp,,Idp,,+l—*/3PDSYM = 16.363... (5.20)

Becausethe good point lies on the dominantsymmetry line, the two points usedto calculate thedistancedo,, arereflectionsof one anotherby thedominantsymmetry.Thesetwo points arethe onesbornacrossthe symmetryline from the bad point of theprecedingorbit.

The observations(iii) and (iv) correspondto self-similar scalingalong and acrossthe dominantsymmetry line (for area-preservingmappings,the scalinghasbeenextendedto otherdistanceswithinthe 2”-cycle in Gunaratneand Feigenbaum[1985]).An alternativewayto definethedistancesdan and

—O~21

__________________________________________-0~430~21 O~I.6

Fig. 5.6. Enlargementof thelower right handcornerof fig. 5.5 whichcontainshalfthepoints of theperioddoublingcascade.Wehavemarkedtherespectivegood pointsof the two-cycle,four-cycleand eight-cyclewith arrows.Thesearethepoints of thecycleon thesymmetry line of H thatbifurcatedalongthisline or, in thecaseof theeight-cycle,will bifurcatealongthis line. Thegoodpointsof successivecyclesalternatebetweenbeingthe leftmost and rightmost points of the two points of the cycle on the symmetry line of H. Thedistancesd~,and d~,are distancesfor theeight-cycleusedto calculatethe scalingsalong andacrossthe symmetryline.

J.A.G. Robertsand G.R.W. Quispel,Chaosand time-reversalsymmetry 141

do,, is in termsof maximal “half distances”[Benettinet al. 1980a,b; Collet et a!. 1981b;van derWeeleet al. 1986]. A half distanceis thedistancebetweena point andthepoint halfway aroundthe periodicorbit from the point. The maximal half distanceis the largesthalf distancearoundthe orbit. Thedistancesda,, anddo,, definedaboveareexamplesof “half distances”for thesymmetricperiodicorbit(we know for instanc~that in any evensymmetriccycle the two pointson thesamesymmetryline arehalfway aroundthe cycle from oneanother).Wecanjustdefined,,,, to bethemaximalhalf distanceonthe orbit anddo,, to be thehalf distanceone quarterof the cycle aroundfrom do,,. This definition is ageneralisationof the definition used in one-dimensionaland higher-dimensionaldissipative perioddoubling (cf. chapter6). Thereis good numerical evidenceto suggestthat the maximal half distancedefinition selectsthedistancesalongand acrossthesymmetryline in anyevent.Notethat themaximalhalf distance(andanyotherdistancebetweenpointsoftheorbit) may notbeunique.Sincea symmetricorbit is its own reflection by an entire family of symmetries,every distancemay occur twice if asymmetrypreservesdistances(e.g., G: x’ = y, y’ = x). The distanced,,,,, will occurelsewherein theorbit if sucha distance-preservingsymmetryexistsand is not dominant(see,e.g.,van der Weeleet al.[1986]).

Thescalings(5.18)—(5.20) arederivedfrom studiesof thesymmetricperioddoublingoftheorigin inexamples2, 3 and 4 of table4.1 for thee valuesshownthere— theexplicit numericalresultsfor example2 are shown in table 5.3. In thesethree examples,the origin turns hyperbolic at C = —1. Thesubsequentsymmetricperioddoubling cascadeis followed to periods1024 in examples2 and3, whicharelargeperturbationsof the area-preservingHénonmapping,and to period256 in example4 which isa small perturbation.The period doubling occurs over a small range of the parameterC withaccumulationvaluesrangingfrom CPD,SYM = — 1.266807. .. in example4 to CPD,SYM = —1.473315.in example3. For thesethreeexamples,we follow the symmetricperioddoublingalongthesymmetryline of their involution H which in all caseswe find to be the dominant symmetry line. This is notsurprising as theseexamplesare directly (example2) or indirectly (examples3, 4) relatedto theMcMillan form of theHénonmapping (1.23) and the simple symmetryG: x’ = y, y’ = x of McMillanmappings(4.14) and (4.15) can be shownnot to be dominant[MacKay1982, section3.1.1]. We tookthe definition of da,, and do,, in termsof distancealong and acrossthe dominantsymmetry line asdefinedin (iii) and(iv) above.Howeverwe checkedthatthe distancedanwasmaximal for theorbit. Inexample2, which hasa distance-preservingsymmetry, this distanceoccurredtwice.

Table 5.4 summarisesin table form the period doubling resultsfor examples2, 3 and 4, and thoseobtained from area-preserving,reversible mappings. It also includes period doubling results forexample1 of table4.1. For this examplethe symmetricfixed point doesnot period-doublebecauseitstraceis boundedfrom belowby 2— = —0.8284...Insteadthesymmetricperioddoublingofoneofits five-cyclesis studied.Sincethis cycle is odd it hasonepointon the symmetryline of H andonepointon the symmetryline of G. It turnsout that thepoint on thesymmetryline of H bifurcatesalongthisline to producea symmetricten-cycle(this point of the five-cycle is insidethe “choppedoff” islandstraddlingthe symmetry line of H on the right hand border of fig. 4.la). Following the continuingcascadeto period 2560 gives similar scalingsto thoseof the fixed points in examples2, 3 and4. Ofcourseif we considerthemappingL~we areagainstudyingthe perioddoubling of a symmetricfixedpoint.

The resultssummarisedin table 5.4 suggestthat symmetricperioddoubling in non area-preservingand area-preservingreversible mappingsbelong to the sameuniversality classes.The numbersareconsistentwith thequadruple-precisionfiguresavailablefor area-preservingmappings(thoseshownintable5.4)andcomparewell with thosefrom double-precisionstudieson them[Benettinet al. 1980a,b;


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+1

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V

I!0

9 ~ ~. ZI’~ ~. Ii

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~ ~

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~Q ~- — .~ 5

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0 111111 I I0.~~~~4

~ 100 N~ ~00~~’Or4~ 5


Table5.4Numericalresults for symmetricperioddoubling in reversiblemappings.

Non measure-preservingreversiblemappings’~ Area-preservingreversible

Example 1 Example2 Example3 Example4 mappings”~

8.7210... 8.7210... 8.7210... 8.72109... 8.721097200...aPOSYM 4.018.. . 4.018... 4.018... 4.018... 4.018076704...$PD.SYM 16.3... 16.37... 16.36.. . 16.3638... 16.363896879...CPDsYM 3.390191684068... —1.273629296599... —1.473315452618... —1.26680794837...

‘~Theseresultsareobtainedfrom double-precisioncalculations.bI Theseresultsaretaken from MacKay [1982,1983a] andare the result of quadruple-precisioncalculations.

Bountis1981]. In non measure-preservingreversiblemappings,the presenceof attractorsandrepellersin otherregionsof the planeduring the rangeof parameterin which the symmetricperioddoublingoccursdoesnot seemto affect the local behaviourbeing quantitatively conservative-like.

The explanationfor the scalings (5.18)—(5.20)in area-preservingreversible mappingsis againintermsof renormalisationgroupanalysis[Colletet al. 1981b;MacKay 1982; Widom andKadanoff1982;Eckmannet al. 1984]. It is shown that in the spaceof thesemappingsthereis a universalmappingwhich is a hyperbolic fixed point under the action of a doubling operator,and whose presencedeterminesthe period-doublingbehaviourin typical one-parameterfamilies of mappings.Again itwould be interestingto seeif this renormalisationgroupargumentcould beextendedto accountfor theresultsgiven here.

6. Dissipativebehaviourin reversiblemappings

In this chapterwe study attractorsand repellersin non measure-preservingreversible mappings.Attractorsand repellersare necessarilyasymmetricfeaturesof reversiblemappingsas pointedout inthegeneraldiscussionofreversiblesystemsin section3.1. Herewewill mainlyconcentrateon attractorsbecausetheir influenceon the dynamicsof the mapping is more transparentthan that of repellers.Attractorsarecommonin dissipativesystemsasdiscussedin section2.1. To investigatethedissipativefeaturesof reversibleplanarmappingsquantitatively,theperioddoublingof attractingasymmetricfixedpoints in someof the examplesof table 4.1 abovehasbeenstudied.

It is very well known from the study of dissipativesystemsthat they too haveperiod doublingcascades like conservative systems(in fact historicallyperioddoublingwasrecognisedfirst in dissipativesystems).Howevertherearesignificantqualitativeandquantitativedifferencesbetweenthecascadesinthe two systems.Perioddoublingof simple attractorsin dissipativemappingsprovidesa universalrouteto chaos[Feigenbaum1980]. The chaoticattractingstructuresthat result often havea self-similar orfractal nature.For mappingsof theplanethecanonicalexampleof this processis Hénon’sdissipativemapping

x’ = —O.3y, y’ = —x+ 2Cy+ 2y2. (6.1)

This orientation-reversingmappinghasJ = —0.3 throughouttheplane.It hastwo fixed pointsandis notreversiblebecausetheeigenvaluesof thesetwo pointsarenot reciprocals(the pointscould only possiblybe asymmetricif themappingwere to be reversiblebecauseJ ~ —1). Hénon[1976]showedthat asthe


mappingparameteris varied,an attractingfixed pointbecomesunstableandbifurcatesto anattractingtwo-cycle which in turn loses stability andbifurcatesto an attractingfour-cycle etc.*) As the mappingparameterC is furthervaried,theperioddoublingprocesscontinues,resultingin a cascadeofinfinitelymany bifurcations with the valuesC~’~at which successivebifurcations occur accumulatingat somevalue. Beyondthis valueHénonfoundan apparentlyaperiodic(strange)attractorwhich hasthenatureof a direct productof a line with a Cantorset.

The successivebifurcation parametervalues ~ for the mapping (6.1) were shown to convergeasymptoticallygeometrically,i.e.,

~ ~ (6.2)

where ~PD,DIss = 4.669... [Derridaet al. 1979]. This value is significantly different to the value forarea-preservingperiod doubling [cf. eq. (5.18)]. This same ~1’D characterisesperiod doubling inone-dimensionalsystems[Feigenbaum1978, 1979],where it hasbeencalculatedto many places,e.g.,8PD,DISS = 4.669201 609 1... It also characterisesperiod doubling in higher-dimensionaldissipativesystems[Colletet al. 1981a].

More generally,the behaviourdescribedaboveis observedin mappingsof theform (6.1) with —0.3replacedby a parameterB, i.e.,

x’By, y’=—x+2Cy+2y2. (6.3)

Whenthevalueof B satisfies B I <1, thesemappingsaredissipative— wecollectivelycall themHénon’sdissipativemapping [notethat when B = 1 (6.3) is the reversiblearea-preservingHénonmappinginMcMillan form]. Sincethe Jacobiandeterminantof themappings(6.3) is constant(J = B) the returnJacobiandeterminantof a 2”-cycle is B2 . Apart from ~PD,DISS,theperioddoublingin thesemappingsisalso characterisedquantitatively by one orbit scaling factor aPDDISS which measuresthe ratio ofmaximalhalf distancesof successiveorbits in theperioddoublingprocess[cf.eq. (5.19)wheredan nowmeansthebiggestdistancewithin the cycle betweena point and the point halfway aroundthe cyclefrom that point]. The limiting valueof this ratio for dissipativesystemsis also significantly different tothat found in conservativesystems(cf. table 6.2 below). Furthermorethere is now no secondorbitscalingfactor /3pD as thereis for conservativesystems.

The universality of the constants6PD,DISS and aPDDJSS(i.e., their appearancein many dissipativemappingsof the plane) hasbeen explainedin termsof the renormalisationgroupand the fact thatlocally arounda fixed point many dissipative mappingsreduceto Hénon’sdissipative mapping.Theapproachof theô~to ~D.DIss in themappings(6.3) asafunctionof thereturnJacobiandeterminantofthe2~-cyclehasalso beenshownto beuniversal[Zisook1981; Quispel1985; vanderWeeleetal. 1985,1986].

In this chapter it is shown that asymmetric fixed points in reversible mappings that are notmeasure-preservingcommonlyperiod-doubleandthat this processrepeatsitself, leadingto a cascadeof2~-cycles.This asymmetricperiod doublingis characterisedby universalscalingexponentsthat arethesameas those foundin dissipativemappings(cf. also Quispel and Roberts[1989]).The approachtotheseexponentsas afunctionof the returnJacobiandeterminantsof the cyclesalso appearsto be the

‘~Actually Hénonstudiedthemappingx’ = y + 1 — ~2 y’ = 0.3x which is simply relatedto (6.1) cf. Note 35 in Helleman[1983].


sameas in dissipativemappings(section 6.1). Furthermore,beyond the accumulationpoint of theperioddoubling,attractorsareobservedthat arevery reminiscentof Hénon’sstrangeattractor(section6.2). Heuristically, theseresultsare expectedbecausethe period doubling of an initially attracting,asymmetricfixed point typically occursin a region of phasespacewhere theJacobiandeterminantissmallerthan 1 (cf. theJacobianplots in chapter4), as in a dissipativemapping.

It is worth rememberingin this chapter that becausethe mappingsstudied are reversible, thepresenceof an attractor implies the presenceof a repeller. Consequentlyevery period doublingsequenceof attractorsin onepartofthe planeis accompaniedby a mirror perioddoublingsequenceofrepellersin anotherpart of theplane,andeverystrangeattractoris accompaniedby a strangerepeller.

Finally, a numerical note about finding asymmetricperiodic orbits. Without the benefits ofsymmetry,a 1D searchalongsymmetrylines can no longerbe conductedasin chapter5. A searchforasymmetricn-cyclesinvolvesa genuinely2D searchfor solutionsof eqs.(2.5).A satisfactorytechniquefor finding thesesolutions is via repeatedapplicationsof a two-dimensionalNewton’smethod(cf. Buck[1978,chapter10]).

6.1. Perioddoubling ofasymmetricfixed points

A fixed point of a mapping of the planecan only be attractingif neitherof the eigenvaluesof itslinearisationare outsidethe unit circle of the complex plane. Genericallyan attractingfixed pointperiod-doubleswhenone of theeigenvaluescrossestheunit circle at —1 and it ceasesto be attracting(cf. section2.1).The reversiblemappingsin table4.1 areorientation-preservingthroughouttheplane.The attracting asymmetricfixed points (x0, y0) of examples1—4 divide into two classes:thosewith aJacobiandeterminantJ = J(x0, y0) suchthat0<J< 1, which typically variesas C is changed,or thosewith J= 1 independentof C. We now presentperioddoubling resultsfor thesetwo classes.

6.1.1. Asymmetricfixedpoints with 0<J< 1Thetypical behaviourof theeigenvaluesA1 and A2 of suchan asymmetricfixed pointprior to, during,

and after period doubling is reasonablywell describedby that of a fixed point in a one-parameterdissipativemappingof the plane(6.3) with J= A1A2 = B, constant,and0<B <1.

A descriptionof the behaviourin this last casewas given, for example,in van der Weele et al.[1986].As a mappingparameterC is varied in onedirection,the cigenvaluesmovefrom both beingonthe positivereal axis to both beingon the negativereal axis via intermediatetraversalof the circle inthe complexplanewith radiusB

112. Theycoincideon the negativerealaxis whenA1 = A2 = —B”

2, andwhenA

1 = —1, A2 = — B the attractingfixed point losesits stability anda two-cycle is born. At this pointof bifurcation the trace of the linearisationof the fixed point, denotedTr~1tsatisfies

Tr~’~= —(1 + B). (6.4)

The eigenvaluesof the two-cycle whenit is createdare,by continuity, 1 and B2. Its returnJacobian

determinantis therefore 2 The above descriptionwith B —* B2 then describesthe motion of theeigenvaluesof the returnJacobianof the two-cycle as C is varied. It bifurcatesto a four-cyclewhenitseigenvaluesare —1 and — B2. More generally,the 2”-cycle bifurcateswhen

Tr~= —(1 + B2~). (6.5)


The returnJacobiandeterminantB2~of the2~-cycledecreaseswith the lengthof the cycle and tendsto0 for n—~oo.

In a mappingwith nonconstantJacobian,the2~-cyclein a period-doublingsequencebifurcatesata C

valuewhen

Tr~= Tr(dL2~)= —[1 + Det(dL2~)J, (6.6)

which is a generalisationof (6.5). The C valueatwhich (6.6) is satisfiedis theonly valueatwhich theimplicit function theoremfails to guaranteethe isolationof the2~-cyclefrom cyclesof twice its length,cf. the discussionabove andbelow (2.11) in section 2.1. Equation(6.6) applies to the asymmetricattractingfixed points and cyclesin reversiblemappings,where the (return) JacobiandeterminantJ,typically changeswith C. Consequentlythe eigenvalues,when they are complex,do not move on acircle but on a deformedcircle whoseradius varies with the polar angle. Becausethe windows ofstability of successivecyclesin theperiod-doublingcascadebecomesmallerandsmaller,the descriptionabovefor constantJ becomesasymptoticallycorrect as2~—~~.

Asymmetricperioddoublinghasbeenfollowed in examples2, 3 and4 of table4.1. In example3, westudy the period doublingof the asymmetricfixed point (~[1— C + \/( 1 — C)2 + 4], —1) which turnsunstablewhen C = 1 — 2\/~= —2.46... At this stageits x coordinateis 2+ \/~= 3.73... and itsJacobiandeterminantis ~. In example4, theperiod-doublingcascadearisingfrom the asymmetricfixedpoint(1 — C, —y

1) is followed. This fixed point turns unstableat C = — 12.943.. . whenits coordinatesare (13.943.. . ,—175.5. . .) and the value of its Jacobiandeterminantis approximately0.02. Theresultsfor example2 arenot obtainedfrom theasymmetricfixed pointsat(0.909. . . , —0.909...) and(—0.909... , 0.909...)becausethe Jacobiandeterminantat thesepoints is always equalto 1 (theirperioddoubling is discussedin section6.1.2 below). Insteadwe study the perioddoublingof anotherasymmetricfixed point P that becomesattractingbelow C = 1/6 = 0.166..., after colliding with thepoint (—0.909. . . , 0.909. . .). This interactionof the two points is depictedin fig. 6.1 and is called atranscriticalbifurcation(cf. GuckenheimerandHolmes[1983,p. 149]; alsoPostet al. [1990a],whogivea discussionof suchbifurcationsin reversibleand nonreversiblemappings).At C = —0.21014.. . , thefixed point Pturns unstableandperioddoubles,with coordinates(—0.826. . . , 0.956. . .) andJacobiandeterminant0.866. . . Tabulationof theperiod-doublingcascadearisingfrom Pis presentedin table6.1[notethat theparametervaluesshownaretheso-calledsuperstablevaluesC~UPdefinedby Tr(dL

2 ) =

01.The critical exponents8PD,ASYM and aPDASYM and accumulationparametervalue CPDASYM that

result from tabulationslike that in table6.1 are shown in the left handcolumnof table 6.2. Table6.2confirms that asymmetricperiod doubling in non measure-preserving,reversiblemappingssharesthesameuniversality classas period doubling in quadraticdissipativemappingsof the planelike (6.1).

We cannotstudy the period doubling of the asymmetricfixed points in example 1 becausethesepoints do not period-double.Howeverwe usethesepoints to illustrate thebehaviourof thesymmetrylinesaroundsimpleattractorsand repellersin reversiblemappings.Symmetrylinescannotintersectinthebasinof attractionof an attractoror thebasinof repulsionof a repeller.This is becausethepoint ofintersection of any two symmetry lines is always a symmetric periodic point and the correspondingsymmetricperiodicorbit cannotbe attractedto something.Example1 providesa good illustration ofthis becausewe havethe two asymmetricfixed pointscloseto thesymmetricfixed pointsoonaftertheyareborn from it. Moreoverasindicatedin figs. 4.laandb, thereappearto be closedcurvesenclosingthese three points. A coarse-grainedsampling of the points within the innermostof thesecurves


0919

_________________________________________________0~899—0~919 —0~899

0919

______________________________________________O~899—0~919 —0~899

Fig. 6.1. Transcriticalbifurcationinvolving theasymmetricfixed point(—0.9090.. . ,Q.9090. ..) in example2 of table4,1. This point, whichhasthesesamecoordinatesandJ = 1 for all C, is at thecentreof eachpicture. It is elliptic for 0.166.. . <C<0.5, in which rangeit is foundto bespirally repelling. In the top picture, C = 0.192 and we show one orbit winding out from it. Numericallywe find anotherfixed point P at(—0.9132...,0.9048...) which has J=1.0093... and is a saddle. In the bottom picture C=0.140. The point P has moved to(—0.9045.. . ,0.9134.. .) andis now attractingwith J = 0.9903.. . , as shownwith oneorbit spiralling into it. Thefixed point at thecentreis nowasaddle(with I = 1). In the interveningrangeof C, thepoint P hasmovedup andto the right, passingthroughthe fixed point at the centreatC = 0.166...,which is when it becomesattractingandthe fixed point at the centreturns hyperbolic.


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J.A.G. Robertsand G.R.W. Quispel, Chaosand time-reversalsymmetry 149

Table 6.2Numericalresultsfor asymmetricperiod doubling in non measure-preservingreversiblemappingsandcomparativeresultsfor period doubling in

dissipativemappings.

Non measure-preservingreversiblemappings~Dissipative

Example2 Example3 Example4 mappings’~

ÔPDASYM 4.669... 4.6692... 4.6692... SPDDISS = 4.6692016091..OPDASYM 2.502... 2.5029... 2.502... aPDDISS 2.5029078750.C~

0ASYM —0.3038681432... —3.17909533968... — 14.4266806188...

I> Theseresultsareobtainedfrom double-precisioncalculations.b) Theseresultsare taken from Feigenbaum[1980]andare theresults for one-dimensionaldissipativesystems.

indicatesthatmostof thepointsappearto be attractedto theattractingfixed pointbelow thesymmetryline of G: y= C/2=1.435.

In fig. 6.2wereproducefig. 4.lb with only thetwo innermost(apparently,closed)curvesshown(thecolour versionof fig. 6.2 appearsat thefront of the report;a blackandwhitecopy is printedhere).Thesymmetry linesof the involutions H and G of example1 are shownin black, anddrawnin colourarethe resultsof iteratinga portion of the symmetry line of G by themapping example1 for two (darkblue), eight (red) and 16 (green) iterations.We also showthe results of iterating this portionof thesymmetryline of G by the inverseof themappingexample1 for two (light blue),eight (brown) and 16

~ 2-135

) /

R / -- -

/ Fig. 6.2. Illustrationof thebehaviourof symmetrylines in the

• vicinity of attractorsandrepellersin reversiblemappings(acolour versionof this figure is found at the beginningof thereport).This pictureis fig. 4.lb, aphaseportrait of exampleIwhenC = 2.87, with theremovalof someof theorbitsshownthere and the addition of somesymmetry lines. The twoinnermostapparentlyclosedcurvesfrom fig. 4.lb are repro-ducedheretogetherwith oneof thesmall islandssurrounding

the symmetriceight-cycle(right-handside). The symmetricA fixed point at thecentreof thepictureis at theintersectionof

the (horizontal)symmetryline of Gand thesymmetryline ofH, both of which aredrawn in black. Two arrowsmark theattractingasymmetricfixed point(A) in thebottomhalfof the

picture andtherepellingasymmetricfixed point (R) in thetop~ ~ /1 half. Thecolouredlines aresymmetrylines createdby iterat-

______ ing the symmetry line of G The coloursdarkblue red andgreencorrespondto two eight and 16 iterationsof this line

~35 underthemappingexample1 Thecolourslight blue brownu andpu~lecorrespondto 2, 8 and 16 iterationsof the line

1~2O65 22065 undertheinverseof example1.


(purple) iterations. It follows from (3.14a) with i = ±2,±8, ±16 that thesecolouredcurvesare alsosymmetry lines for the mapping example 1. Moreover Fix(L2’ 0 G) = G {Fix(L -2t o G)}, which isobvious from the symmetryof thepositive andnegativeiteratesvia reflectionin the line y= C/2. It isimmediatelyapparentthat thesesymmetry lines appearto havehardly any intersectionswithin theinnermostcurve but an abundanceof intersectionsoutsideof it (the rightmost intersectionof the redline with thehorizontalsymmetryline of G resultsin a periodicpointof period8 encircledby the smallislandshown).Oneobviouspoint of intersectioncontainedwithin the innermostcurvethat cannotbeinthebasinof attractionis thesymmetricfixed pointwhich “anchors”thesymmetrylinesasthey must allpassthrough it.

The behaviourof thesymmetrylines within the innermostcurve is consistentwith theirbeingin thebasin of attraction.The second(dark blue) iterateof thepartof thehorizontalsymmetryline of G tothe left of the symmetricfixed point is bent anticlockwiseto avoid the partsof Fix(G) andFix(H).Successiveiterations(red, greencurves)of thepart of the seconditeratewithin the innermostclosedcurve movecloserto theattractingfixed point andwraparoundit, nestingthemselvesaccordingto thenumberof their iteration.Thus although the points of an attractorcannotlie arbitrarily closeto anygiven symmetryline, becausethenthe attractorwould be symmetric,a sequenceof symmetrylines canconvergeto it. In this way,somesymmetryline canbe foundwithin an arbitrarilysmall neighbourhoodof theattractorbut a givensymmetryline cannot.Moreover,becauseof reversibility, if yEFix(G) andthe sequenceL~yconvergesto an attractorA, then GL0y= L~’Gy= L~y convergesto GA, arepeller. Consequently,the points on the part of a symmetryline within a basin of attractionof anattractor in a reversible mapping are also in the basin of repulsion of the repeller, that is, theasymmetricpartnerof theattractor.Thesepointscomearbitrarilycloseto therepellerunderbackwardsiterationand arbitrarily close to the attractorunderforwards iteration. In fig. 6.2, for example,thepointson the purpleline closeto the repellerR iterateto pointson thegreenline neartheattractorAafter32 iterations.Thus the trajectoriesof thesepoints link the attractorsand repellersin reversiblemappings.It seemshighly unlikely thoughthat thebasinsof attractionandbasinsof repulsionareoneandthesameset.

6.1.2. Asymmetricfixedpoints with J= 1In example2 of table4.1 (e = 1.1) thepair of asymmetricfixed points (0.909... ,—0.909...) and

(— 0.909.. - , 0.909. . .) haveJ = 1, independentlyof C. In the rangeof C whenthesepointsareelliptic,it is found numerically that the first point is spirally attractingas indicatedin fig. 4.2b, and that thesecondpoint is spirally repelling.Becausethemodulusof theeigenvaluesofthesefixed pointsequals1,the attraction(repulsion)of thesepoints is weak andconvergenceto (divergencefrom) thesepoints isslowerthecloserone is to them. Suchattractorsandrepellersarenotusuallydiscussedin thecontextofdissipativeor expansivesystemsbecausetypically thesesystemsare takenby definition to have I JI < 1or J~> 1 everywhere.

More generally,considerexample2 oftable4.1 for arbitrarys. The fixed point (1/g, —1/c) hasJ = 1for all C andall c and,for any c, is an elliptic fixed pointwhenC is in therange~ <C < ~. Whenit isan elliptic fixed point, it is stablein the linear approximation.Since J= 1, we can say that it ismeasure-preservingto first order. SinceI = 1, it also passesthe first order test for possibly being asymmetricfixed point (in fact the Jacobiandeterminantaroundthis point is similar to that aroundasymmetricfixed point, varying from 1< 1 on oneside of the point to 1> 1 on theotherside,asin fig.4.ic, cf. also fig. 4.2c). To analytically seewhy (1/ e, —1 Is) of example2 is attracting,we use higherorder (i.e., nonlinear)stability analysisas describedin appendixB. We calculate the quantity G

1,

J.A.G.Robertsand G.R.W. Quispel, Chaosand time-reversalsymmetry 151

involving the coefficientsof secondand third ordertermsin the expansionaroundthepoint to obtain

= 2[s(2C — 1)(8C— 1)(6C— 1)2]! . (6.7)

SinceG1 <0 when~ <C < ~andsis positive, the fixed point (1/c, —1/c) is anattractingfixed point. Ingeneral,G1~ 0 for a fixed point with I = 1 violates third order local measurepreservationandlocalreversibility conditions (cf. appendixB), i.e., thepoint fails to be measure-preservingor symmetricatthe third order.

It is foundthat whenan asymmetricattractingfixed pointwith I = 1 period-doubles,its offspringhasa returnJacobiandeterminantunequalto 1 and theensuingcascadebelongsto the dissipativeregimediscussedin section6.1.1 above.This is to be expectedbecause,without the benefitsof the fixed pointbeingsymmetricor themapping beingmeasure-preserving,an asymmetricfixed point with I = 1 in areversiblenon measure-preservingmappingcannottypically sustaina conservativeJacobiandetermin-ant throughoutthecyclesof its period-doublingcascade.

Thus in example 2, when C = 0.5 and the attracting fixed point (0.909...,—0.909. .1) turnshyperbolicwith both eigenvaluesat —1, the two-cycleborn from it quickly assumesa returnJacobiandeterminantvalueslightly lessthan1 (e.g.,0.9993... atC = 0.502)andis attracting.Thebehaviourofthe eigenvaluesof the 2”-cycles, n � 1, of the period-doublingcascadeis qualitatively similar to thebehaviourof the eigenvaluesof the fixed points and cycleswith 0< 1< 1 describedabove.Howeverbecausethe return Jacobiandeterminantof the two-cycle is so close to 1, the decreasein the returnJacobiandeterminantof later cyclesis slow (the value for the2”~~

1-cycle is approximatelythesquareofthe value for the 2”-cycle). Associatedwith this is a slow decreasein thesuccessive15C valuesin theperiod doublingcascade[definedin (6.2)]. Despitethis slownessit seemsthat the perioddoubling isagaincharacterisedby the dissipative exponents,e.g., the &.~. valueformed from the 1024, 2048 and4096-cyclesis 4.678. -. and the a,, value calculatedfrom the 1024 and 2048-cyclesis 2.517... Onewould expectto see moreaccurateconvergenceto the dissipativevaluesif it were possibleto obtaingood numericalresultsfor enoughperiodsof the cascade.

In fig. 6.3, the decreaseof the 6,,~. in example2 is illustrated by plotting their valuesagainstthereturnJacobiandeterminantsof the2”-cycle evaluatedat superstability.The 2”-cycle is themiddle cycleof the 2” ~, 2”-, 2” +‘-cycle triplet whose superstableC~valuesare usedto calculateô~..The datapoints from example2 are supplementedby those from anotherreversible non measure-preservingmappingwith an asymmetricfixed point with I = 1 (cf. example1 in Quispel andRoberts[1988];thefixed point (0, —20v’~)of this examplegives rise to a “dissipative” period-doublingcascade).

In dissipativemappingsof theplanewith constantJacobiandeterminantI = B < 1, thevaluesof &.~.

in the period-doublingprocessregardedas a function of theeffectiveJacobianBe B2 (which is thereturn Jacobiandeterminantof the 2”-cycle) fall on a universalcurve. This is due to the relation

~~(B2), which holds increasinglywell asB—*0. Thereforenotonly arethevaluesof 6PD forthe conservative(B = 1) case and the dissipative (B < 1) case universal numbers,but for weakdissipation(i.e., B ~s1) the successivevaluesof ô~ descend(“crossover”) from the first to thesecondwith increasingn in a universal fashion. This follows from numerical results and renormalisationarguments[Zisook1981; Quispel1985; vanderWeeleet al. 1985, 1986].To illustrate this crossover,theuniversalcrossovercurve for theHénondissipativemapping(6.3) is~also plottedon fig. 6.3. This curveis obtainedfrom thebestfit to ~ versusBe B2 datapointsfor theperioddoublingof the fixed pointat theorigin in (6.3) for B = 0.99 andB = 0.5. Theconclusionfrom ~g.6.3 is that theperioddoublingof I = 1 asymmetricfixed points in reversible mappingsfollows the universalcrossoverbehaviour

152 l.A.G. Robertsand G.R.W.Quispel, Chaosand time-reversalsymmetry

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Return Jacobian Determinant

Fig. 6.3. Crossover-typebehaviourin the perioddoubling of I = 1 asymmetricfixed points in non measure-preservingreversiblemappings.Shownarethe for theperioddoublingcascadearisingfrom sucha point in example2 (1~),plottedasa functionof thereturnJacobiandeterminantofthe2’-cycle. Also plotted aredatapoints (I) from example1 of QuispelandRoberts[1988].Thecurvedrawnon thefigure is thecrossovercurvefor theHénondissipativemapping.The arrow on the left handverticalaxismarksthedissipative

8PD = 4.6692... , andthearrowon theright handvertical axismarks theconservative8pD = 8.7210.

previouslyfound in the dissipativemappings(6.3) (note that the pointswhere B 1 arenot terriblysignificantas theuniversality holds for B —+0). This agreementmight be expectedbecauselocally thereturnJacobiandeterminantof cyclesbecomesconstantin C ashigherandhighercyclesareconsidered.

Although thecurvefor theHénondissipativemappingin fig. 6.3 appearsto decreasemonotonically,it hasin factbeenshownto havea dip at a small valueof B ~ whereömin 4.6635 [Quispel1985;van der Weeleet al. 1985, 1986]. For theI = 1 asymmetricfixed points it is not possibleto numericallyfollow theperiod-doublingcascadeuntil periodicorbits havenegligible Jacobiandeterminants.How-everthis is possiblefor the fixed pointswith 0<I < 1 discussedin section6.1.1 above.In thesecases,the period-doublingdata each show a decreasein the &.~ to at leastone value below 8PD.DISS =

4.6692...and thenan approachto this valuefrom below, asshownfor examplein table6.1.

6.2. Strange attractors

The combined results of sections5.2 and 6.1 suggestthat conservativeand dissipative period-doubling processescan occurin different parts of the planein a one-parameterreversiblemapping.They are identified as such by their scalingexponents8PD and aPD, and ~D (for symmetric perioddoubling). For the examplesin table 4.1, the rangesin parameterin which thesetwo period-doublingprocessesoccuroverlapto varying degrees.

In example4 of table 4.1, for instance,the point (1 — C, —y1) = (2.266,—1102.132...)is “just”

attractingat C = —1.266— its eigenvaluesare 0.9907and 5.188X i0~.At this C valuethe symmetric

IA. G. Roberts and G.R.W. Quispel,Chaosand time-reversalsymmetry 153

perioddoublinghasreacheda stagewhereanelliptic 8-cycle existsabouttheorigin. As C is decreasedthe eigenvaluesof the point (1 — C, —y1) move towardsthe (superstable)point at C = —9.950...wheretheir sum equals0. This is well beyondtheaccumulationpoint of thesymmetricperioddoublingat CPDSYM = —1.266807...Thus the symmetric period doubling accumulatesat a parametervaluesuchthat theasymmetricperiod doublingof theattractingpoint(1 — C, —y,) hasnot evenbegun.Thesamebehaviourarisesin example3 — theparameterintervalsfor theexistenceof stableasymmetricandsymmetriccycles in their respectiveperiod-doublingcascadesare shownin fig. 6.4a.

In example2, the situation is reversed.Here the parametervaluesfor the period-doublingcascadearisingfrom theasymmetricfixed point F, which becomesattractingafter it passesthrough the point(—0.909. .. , 0.909. . .) at C = ~, accumulateat CPDASYM = —0.303 8681. .., which is well within therange —1 < C < 1 in which the origin is elliptic and surroundedby KAM curves— seefig. 6.4b. Theorigin doesnot symmetricallyperiod-doubleuntil C = —1. Notethat theperiod-doublingcascadeof thefixed point (—0.909..., 0.909...) also accumulateswell within the range —1 < C < 1 although itproceedswith increasingC afterthis fixed point period-doublesatC = 0.5. The parameterwindowsforthis period-doublingcascadeare not shownin fig. 6.4b.

In dissipativesystems,perioddoubling canaccumulatewith theappearanceof an aperiodicattractor(the “FeigenbaumScenario”,cf. Eckmann[1981]).This attractortypically hasmanypieceswhich oftenrecombinevia “inverse bifurcations” as a parameteris varied (e.g., Schmidt [1988]).The aperiodicattractoroften disappearsfor denumerablymanywindows in the parameter,being replacedby anattracting periodic orbit which again period-doubles,etc. In the Hénon dissipative mapping (6.3)aperiodic attractorsand their evolution havebeenwell studied(cf. e.g., Hénon [1976],Simó [1979],Feit [1978]and Cvitanovic et al. [1988]).The attractorsare describedas being “strange” by mostauthorsbecausethemotion inducedon themis chaotic;in additionthey possessself-similarstructureonall scales.Very recently, it hasbeenanalytically proven that the mapping (6.3) doesindeedhaveastrangeattractorfor a rangeof parametervaluesB and C (cf. Benedicksand Carleson[1989,1991]).However,this rangecoversonly very small valuesof B anddoesnot includethevalueB = —0.3 studiedby Hénon[1976]and manysubsequentworkers.

In all casesof period doubling of asymmetric(0< 1< 1) fixed points that we have studied inreversible mappingswe haveobservedthe formation of an apparentlyaperiodicattractor for manyparametervalues beyond CPDASYM.*) Consider, for instance, example 3 for which CPDASYM =

(a) (b)asymmetric attractingcycles asymmetricattractingcycles

42 1 421II I p III I

—3 —2 —1 0 1 —1.5 —1.0 —0.5 0.0 0.5 1.01.1 I I I I I i~C I 1 I I I ~I I I I P—C

cPD5VM CPDASYM

II I I II I I42 1 42 1

symmetricelliptic cycles symmetricellipticcyclesFig. 6.4. Intervals in the parameterC for theexistenceof (asymmetric)attractingcyclesand (symmetric)elliptic cyclesin (a) example3, and(b)example2. Thenumbers1, 2 and4 label thewindowsof existenceof cycleswith theseperiodsin the asymmetricandsymmetricperioddoubling.

*1 Suchanattractormay alsoarisein thecaseof asymmetric (J = I) perioddoublingbuttheslowerconvergenceoftheC0 makesit moredifficult

to investigate.


—3.179095... We identify a four-piece aperiodicattractorat C = —3.19. Points attractedto it cyclearoundthe four pieces.Thesefour piecesmergeto give two piecesfor —3.23<C < —3.20 andonepiece for —3.301� C s —3.23. The attractordoesnot appearto exist for C= —3.302. Like Hénon’sattractor,self-similarity and an apparentlyendlessseriesof folds are seen in this attractor.This isillustratedin fig. 6.5 whereweshow theattractorin 6.5aandsomemagnificationsofit in 6.5b—d.In fig.6.6 a further illustration is providedof the behaviourof symmetrylines aroundattractors.Here theattractorhasmorestructureand“occupies”morespacethantheattractingfixed point in fig. 6.2. Againit is foundthat successiveiterationsof a partof a symmetryline in thebasinof attractioncomeclosertothe attractor.In the presentcase,they fold over to mimic theshapeof the attractor.In the basin ofattraction, the iterationsare nestedparallel to one anotherand to the attractorso that they do notintersecteachotheror cut the attractoritself.

Similar qualitativeresultsarefoundin examples4 and2. For example4 at C = —14.8 theaperiodicattractor is a large parabolicshaped“curve”. Its basin of attraction is very large: points that areattractedto it include (—5,1), (30, 3) and(0, 6.798).It appearsto bethemain dynamicalfeatureof themapping,aspointsnot attractedto it apparentlyescapeto infinity. Theshapeof this attractoris againreminiscentof that in Hénon’sdissipativemapping.

The aperiodicattractorobtainedin example2 beyondCPDASYM = —0.303868 1 ... compriseseightsmall pieces and has a very small basin of attraction. Significantly the attractor is not the maindynamicalfeatureof the mapping. It coexistsover the range of C in which it exists with the KAMregionaroundtheorigin, which is an elliptic fixed point. This is shownin fig. 6.7 for C= —0.303 9785.Thereforethe mappingin this rangeof C hastwo types of boundedmotion: themotionon andinsidethe regular region formed by the KAM curves around (0,0); and the motion towardsand on theattractor.Figure6.8 shows in moredetail someofthepiecesof theattractor.Themagnificationof anyoneof the piecesis againsuggestiveof fractal naturelike fig. 6.5. The attractorseeminglyceasestoexistfor C a little lessthanC= —0.3039785(e.g.,the initial conditionusedto generatefig. 6.8 escapesto infinity at C = —0.303979 0), and at destructionit still seemsto compriseeight pieces(note thatattractorswith piecesarisein manyphysicalexamples,e.g., Short and Yorke [1984]).

• It appearsthat the aperiodicattractorsfoundin thesenon measure-preservingreversiblemappingsaregoodcandidatesfor beinglabelledstrangeattractorsa la Hénon— attractorson which thedynamicsis chaoticand that havea geometricfractal nature.Qualitatively the picturesin fig. 6.5 suggestthegeometricproperty;watchingthe generationof pointson the attractoron acomputerscreensuggeststhe dynamicalproperty.Quantitatively,themotion on theattractorsin examples2 and3 is verified tobe chaoticfrom theexistenceof a positiveLyapunovexponentj.i~.We calculateboth exponentsp.~and~ for eachof the examplesaccording to the standardprocedure(cf. Eckmannand Ruelle [1985],LichtenbergandLieberman[1983],Mayer-Kress[1986],Young [1983]).That is, we follow an initiallyorthonormalpair of vectors around the attractorand at each iteration of the mapping, apply thelinearisedmapping (the Jacobianmatrix) to thevector pair and re-orthonormalisethem. The lengthrescalingfactorsx1 (i) for the first vectorandx2(i) for thesecondvectorat the ith iterationarecollectedtogetherto define ~ and~ via = lim[(1/n) E~log x1(i)], for j = 1, 2, wherethe limit is asn—~°~. Itis foundthat ~ 0.38and /~~2 —0.98 for theattractorof example3 depictedin figs. 6.5—6.6.The factthatp.~is positive indicatesexponentialdivergenceof initially closepointson theattractor;theaveragearea change per iteration is by a factor exp( p.~+ ~) 0.55. Fortheattractorof example2 depictedinfigs. 6.7—6.8, the resultsare 0.05 and p2 —0.23 so that exp(p1 + ~) 0.84. The averageareareductions in both casesemphasiseagainthesimilarity in theseregionsof the phasespacebetweenthereversiblemapping and a dissipativemapping.


-030

—170382 5~22

(a)

\\~ -090

\

\\ .~‘

\ \\.

\.‘

—1•104-42 (b) 462

Fig. 6.5. Aperiodicattractorin the mappingexample3 for C = —3.300,and successive_magnifications[(b) magnificationof the boxedregion in a,

etc.]. At the centre of each picture is the hyperbolic fixed point ((1— C + \f(1 — C)2 + 4)/2, —1) = (4.52118. . . , —1) which was originally

attracting.Theattractoris generatedby iteratingthepoint(4.5, —1.1),for up to 5 million iterationsin thecaseof d, without plotting the first 2000iterates.

156 JAG. Robertsand G.R.W. Quispel,Chaosand time-reversalsymmetry

.‘~-. \ ~O~99O

~

.~\ \\

.~

~ ~\\ \

~\•‘\\ \\

\ ~\\ \~‘ \ -1010

4~511 (c) 4~531

-09990

‘C.. \

..,. \.

‘~ ..‘-

.~.. ‘\\

\\

“. —100104~52O2 (d) 4~5222

Fig. 6.5 (cont.).


~. -0~990..~ \\\

~\ \\

‘\ ~

\\ ‘\

-1 0104511 4531

Fig. 6.6. Figure6.5c with theadditionof somepartsof thesymmetrylines of L’G (single arrow)andL7G (doublearrows),whereL is themapping

example3. Theselines areobtainedby taking the secondandthird iteratesunderL of thesymmetryline of H for example3.

.1,~,

-1•1 ~ T:~.. :

—1~1

Fig. 6.7. Phaseportrait ofexample 2 when C= —0.3039785.NotetheKAM regionaroundtheoriginandtheboxedeight-pieceaperiodicattractorin thetop left corner. Thesymmetryline of H is also shownbut thesymmetry line of G, y= x, is omitted.


0917N

‘-S

0~883—0~737 —0~703

O~889o

O~8876—O~7O88 —O~7O58

Fig. 6.8. Enlargementof theeight-pieceattractorof fig. 6.7. Thetoppictureshowsthefourrightmostpiecesof theattractorandthebottompictureshowsthetwo rightmostpiecesof thetoppicture.Theattractorisgeneratedby iteratingthepoint(—0.883039,1.01843)20000timesandplottingonly the last 10 000 iterates.

J.A.G. Robertsand G.R.W. Quispel,Chaos and time-reversalsymmetry 159

7. Interplayof conservativeanddissipativefeaturesof reversiblemappings;a symmetry-breakingbifurcation

Thecombinedresultsof theprecedingtwo chaptersindicatethat nonmeasure-preserving,reversiblemappingsprovidea uniqueopportunityto study the interactionof conservativeanddissipativefeatures(e.g., KAM curves and attractors)within the one mapping, becauseof the identification of thesefeatureswith, respectively,symmetricandasymmetricsets.In this short chapter,weconcentrateon oneaspectof this interplay, namelya symmetry-breakingbifurcation in which two asymmetricperiodicorbits bifurcate from a symmetricconservative-likeperiodicorbit.

This symmetry-breakingbifurcation is seenimmediately at the level of fixed points in example1 oftable4.1,whentwo asymmetricfixed points,one attractingwith I < 1 andonerepellingwith 1> 1, areborn from the symmetricfixed point at parametervalue C = ±2\/~.This bifurcation hasalso beenobservedin othernon measure-preserving,reversiblemappingsby Politi et al. [1985,1986a,b], Bullett[1988],Roberts[1990a]and Postet al. [1990a](thelatter referencealso discussesotherbifurcationsinreversiblemappingsof the plane). The symmetry-breakingbifurcation is shown schematicallyin fig.7.la, with another possibility in fig. 7.lb. The analysisin Post et al. [1990a]suggeststhat suchsymmetry-breakingbifurcationsshould occur quite frequently in non measure-preserving,reversiblemappings.In other(nonreversible)mappingssimilar bifurcationsof two fixed pointswith, respectively,I < 1 and1> 1 from a fixed point with I = 1 areexpectedif a secondordercondition on themappingexpansionis satisfied.This condition is preciselythesecondorderreversibility condition“case2b” of

(a) A ~<‘) (a) C

c___ c___

(b) S ~ (b) S

Fig. 7.1. Schematicrepresentationsof symmetry-breakingbifurca- Fig. 7.2. Schematicrepresentationsof symmetry-breakingbifurca-lions in a non measure-preservingreversible mapping: (a) a symmetric tions in an area-preserving reversible mapping. Now all fixed pointsfixed point with I = 1 (horizontalline) changesfrom beingelliptic (a have I = 1. (a) asymmetricfixed point changesfrom a centreto acentreC) to hyperbolic(a saddle5) asthemapparameteris varied, saddleandemits two asymmetriccentres;(b) asymmetricfixed pointspawningan asymmetricattractingfixed point A with I < 1 and an changesfrom asaddleto acentreand emits two asymmetricsaddles.asymmetricrepelling fixed point R with 1>1; (b) a symmetricfixedpointwith I = 1 changesfrom beingasaddleS to acentreC andemitstwo saddles,one with 1>1 and one with J <1.

160 JAG. Robertsand G.R.W. Quispel, Chaosand time-reversalsymmetry

table 3.2; the othersecondorder reversibility condition at “case2a” of table 3.2, which is also thesecondorder condition for local measurepreservation(cf. “case 2”, table 2.1), neednot be, andtypically is not, satisfied.

In one-parameter,area-preserving,reversiblemappings,asymmetry-breakingbifurcationwasfoundby Rimmer [1978,1983]. In this casethere cannotbe attractingand repelling fixed points andI = 1everywhere.It is found that therecan be two possibilitieswhena symmetricfixed point hasrepeatedeigenvalue+ 1 andtheparameteris varied: (a) thesymmetricfixed point changesfrom beingacentretoa saddle;and two asymmetricfixed points, which arecentreseachwith thesameeigenvalues,areborn(fig. 7.2a); or (b) the symmetric fixed point changesfrom a saddle into a centre and emits twoasymmetricsaddleswith identical eigenvalues(fig. 7.2b). The bifurcations in fig. 7. la, b can beregardedas non measure-preservingversionsof the Rimmerbifurcationsin fig. 7.2a, b.

Rimmer showed that the symmetry-breakingbifurcation is a generic one for area-preservingreversible mappings. Previously, Meyer [1970]had consideredthe generic bifurcations in area-preservingmappings— in this wider classof mappings,suchpitchfork bifurcationsasin fig. 7.2 arenotgeneric. MacKay [1982]showedthat the area-preservingand reversible Hénon mapping first hasasymmetriccycles at the level of six-cycles when two asymmetricsix-cycles arise via a Rimmerbifurcationfrom a symmetricsix-cycle. SchmidtandWang [1983]find the bifurcation depictedin fig.7.2ain their study of possiblebifurcation scenariosin mappingsof de Vogelaereform (cf. table 3.1).Beerens[1990]and Post et al. [1990b]have followed the period doubling of the asymmetriccyclesproducedby a Rimmerbifurcation in anarea-preservingmappingand foundthescaling8PD,ASYM to be8.721...De Aguiar et al. [1987]give someexamplesof symmetry-breakingbifurcationsin thevein ofRimmerin the two-dimensionalsurfaceof sectionof Hamiltoniansystemswith time-reversalsymmetry(cf. also the discussionin Ozorio de Almeida and de Aguiar [1990]).Kook and Meiss [1989]discussRimmerbifurcations in four-dimensionalsymplecticreversiblemappings.

Although the non measure-preserving,symmetry-breakingbifurcation is not seenamongthe otherexamplesof table4.1 at the level of their fixed points,it appearsoften in theseotherexamplesif welook at n-cycles,n > 1 (seefig. 7.3). In particular,thesymmetry-breakingbifurcationsoftenoccurso asto interruptthesymmetricperioddoublingprocess.Forexample,thesymmetricfour-cycleshownin fig.7.3 arisesfrom two successiveperioddoublingsof thesymmetricfixed point at theorigin in example3(notethe differentr valueto that shownin the lastcolumn of table4.1 andusedin section5.2).As theparameterC is lowered,the traceof the returnJacobianof this four-cycle decreasesfrom thevalue +2which it initially haswhen it is born from its parentsymmetrictwo-cycle. However,the trace “turnsaround” at a value 1.88 when C = —1.6090...and thenincreasesthrough + 2, which is when thesymmetry-breakingbifurcation occurs.

For sometime after the symmetry-breakingbifurcation in example1 of table 4.1, the threefixedpoints areapparentlyenclosedby closedcurveswhich aresimilar to theKAM curvessurroundingtheelliptic symmetricfixed point before it bifurcates(cf. figs. 4. la,b). A similar situationis seenwith thefour-cyclesin fig. 7.3. We nowinvestigatewhetherthecurvesseenin figs. 4. la, b and7.3 areactuallyclosed.This is suggestedby numericalevidenceaswe do not seethe curvesfattenor degenerateafterfollowing them for many iterations. This contrastswith someexamplesof mappingsin Post et al.[1990a]in which reversibilityis only satisfiedlocally to someorderarounda fixed point (cf. section3.3)and in which it is shownthat theKAM tori changeinto slowly spiralling curvesenclosingtheattractorand repeller (heuristically, the higher the degreeof local reversibility, the slower the discernabledeparturefrom a closedcurveto a slow spiral).

Sinceexample1 is a fairly simple mappingthat is reversibleto all ordersand thesymmetry-breaking

JAG. Robertsand G.R.W. Quispel, Chaosand time-reversalsymmetry 161

-112

11’ ~~:?~

/ -1~26—0~59 —0~45

Fig. 7.3. Phaseportrait of example3 with a = 0.252whenC = —1.64755. A symmetry-breakingbifurcationhasoccurredat C= —1.64459...Atthecentreof thepictureis ahyperbolicpoint of asymmetricfour-cyclelying on amonotonicpartof thesymmetrylineof H whichis alsoshown.Above and below the point of the symmetric four-cycle are, respectively,one point of a repelling four-cycle and one point of an attractingfour-cycle. A trajectorythat winds out from the repellerto the attractoris shown.

bifurcationinvolvesfixed points,we will studyit. We have lookedfor acurve betweenthesymmetricseven-cycleand thesymmetriceight-cycleshownin fig. 4. la. The rotationnumbersof thesecyclesarerespectively1/7 and1/8, i.e., it requiresoneanticlockwiserevolutionaroundthesymmetricfixed pointto visit everypoint ofeachcycle. In section5.1.2 it wasmentionedthat in area-preservingmappingsthelast KAM curve expectedto breakup betweencycles with neighbouringrotation numbersplq andp’Iq’ hasthe rotation number(cf. alsoMacKay andStark [1991]),

(p + yp’)I(q+ yq’), y=(~+ 1)/2. (7.1)

We investigatetheexistenceof a curvewith rotation number

(1+y)I(7+8y), (7.2)

using the residuebehaviourof the symmetric periodic orbits with rotation numbersapproximating(7.2), asdescribedin section5.1.2.

The rationalapproximantsto (7.2),which is noblewith CFE [7,1, 1, 1,.. .], areeasilyobtainedby aFareytreeconstruction[cf. eqs.(5.9)—(5.10)]. The first few approximantsare

fl 1 2 3 5l7~ 8~15~23~38 .

Later approximantsarefoundfrom table7.1 (givenby revolution/period)which lists the intersectionsof the residuecurvesof successiveapproximantperiodicorbits. We areableto locate these(positive-residue)approximantsall on the symmetryline y = C/2 of the involution G for example1, providing

162 l.A.G. Robertsand G.R. W. Quispel,Chaosand time-reversalsymmetry

Table 7.1Intersectionsof residuecurvesof periodicorbits aroundthesymmetricfixed point of example 1 of table 4.1. The parameter value C~”shouldgive

agreementbetweentheresiduesof eachpair of orbits shownto at leastthenumberof figures in R”.

Revo-Period lution Coordinate~ Sym~ C~ R~

t

61 8 0.3467850476567... 199 13 0.345482991 541 ... 1 2.877593007458568 0.2384626745...

99 13 0.344204379987... 1160 21 0.344623077653... 1 2.877959592569219 —2.791032... 0.2571335003... —1.6888...

160 21 0.345083619910... 1259 34 0.344946020861 ... 1 2.877828248670725 —2.767324... 0.246078025... —1.7151

259 34 0.3447800708... 1419 55 0.34482473283... 1 2.877875711077100 —2.715285... 0.2525241... —1.6594...

419 55 0.34488594226... 1678 89 0.34487134038... 1 2.877858231362837 —2.703593... 0.2486396... —1.6658...

678 89 0.34484871857... 11097 144 0.34485347206... 1 2.877864696727028 —2.68586... 0.2509714... —1.6462...

1097 144 0.344861898219... 11775 233 0.34486034678... 1 2.877 862289543 272 0.249555...

‘~“Sym” = 1 correspondsto y = C/2 and “coordinate”is thex coordinateon this line.

furtherevidencethat the “dominantreverserconjecture”for reversiblearea-preservingmappingsholdsin more generalreversiblemappings(cf. our discussionin section5.1.2).

From table7.1, the following critical exponentsfor the noblecurve with rotationnumber(7.2) arecalculated,using superconvergencein the caseof RKAM: RKAM = 0.25009...,8R,KAM = —1.6...,8C,KAM = —2.6... We also find CKAM =2.8778629... Theseresultsagreewell with thosetabulatedintable 5.2 for other non measure-preserving,reversiblemappingsas well as thosefor area-preserving,reversiblemappings(consideringthat we do not continueto very long approximantorbits asin section5.1.2).

The residuesof the approximatingorbits to (7.2) increasewith increasingparameterC, becomingunboundedfor C> CKAM. For C< CKAM they tendto 0 from above.In area-preservingmappings,thebehaviourof the residueshasbeenusedas a practical test for deciding whetherthere are invariantcirclesof a mappingin an interval of rotationnumberspecifiedby two periodicorbits, cf. Greeneetal.[1986].Assumingthat themost robustcircle in the interval is a noble(cf. MacKay and Stark[1991]),onechecksthe residueof periodic orbits at theends of the interval. If the residuesaresignificantlylargerthan 0.25 (0.25 is approximatelyRKAM), then oneexpectsthereis no noblecircle betweentheperiodicorbits andhenceno invariantcircles betweenthem; if the residuesarebetween0 and0.25, oneexpectsthat thereareKAM circlesbetweenthem. If the residuecriterionorconjecturesummarisedby(5.12a)—(5.12c) is assumedto hold in thepresentcase,we havethat for C< CKAM thereexistsa smoothinvariantcircle with rotation number(7.2),whereasfor C> CKAM theredoesnot. As pointedout insection5.1.2,webelievethat, basedon our numericalevidence,the residueconjecturedoeshold. As aresult,it would seemthat therecanbeclosedinvariantcurvesencirclingsymmetric,hyperbolicperiodicorbits andtheir attractingandrepelling offspring after a symmetry-breakingbifurcation.

JAG. Robertsand G.R.W.Quispel, Chaosand time-reversalsymmetry 163

8. Concludingremarksandfuture directions

Let us briefly summarisethework alreadypresentedandconcludewith somepossibledirectionsforfuture research.In this report we havereviewedthe dynamicsof systemspossessingtime-reversalinvariance.We havenot restrictedthediscussionto conservativereversiblesystems,andso havebeenable to compare and contrast a general reversible system with a conservativesystem. We haveconcentratedin particularon reversiblemappingsof the plane. We havereviewedthe propertiesofthesemappingsin broadtermsin section3.1 indicatingthat, in theirfull generality,reversiblemappingscombinethepropertiesof conservativeanddissipativemappingswhich were reviewedin chapter2. Wehavediscussedhowto createreversiblemappingsin section3.2 andtalkedabouttheproblemof testingarbitrary mappingsfor reversibility in section 3.3. Someconservativenonreversiblemappingsweregiven in section3.4.

Chapter4 presentedexamplesof nonconservativereversiblemappings.Thesemappingscan haveattractorsand repellers.We gavesystematicmethodsfor creatinglarge classesof suchmappings.Weusedtheseexamplesto surveysomeof the featuresof nonmeasure-preserving,reversiblemappingsinchapters5, 6 and 7. Chapter5 hasshownthat if we takea non measure-preserving,reversiblemappingand study phenomenaassociatedwith its symmetricperiodicorbits, we find behaviourreminiscentofconservativereversible systems, which is reflected quantitatively in the critical exponentsfor thebreakupof KAM circles andsymmetricperioddoubling.Chapter6 hasshownthat if we takethesamemappingand studyphenomenaassociatedwith asymmetricperiodicorbitswe find behaviourpreviouslyassociatedwith dissipative systems, which is reflected quantitatively in the critical exponentsforasymmetricperioddoubling.As remarkedin section6.2 andshownin fig. 6.4, the two different perioddoublingprocessescan occurat thesametime in differentpartsof theplane.Theseresultsexpandeachof the universalityclassesof mappingswith dissipativecritical exponentsandmappingswith conserva-tive exponents,at the sametime showingthat the two sets of exponentscan be foundin one and thesame(reversible)mapping.Finally, chapter7 illustratessomeof the interplaybetweentheconservativeanddissipativefeaturesof reversiblemappings.

The reporthasshownthat reversibledynamicalsystemscanqualitatively andquantitatively “bridgethe gap” betweenthe traditional “textbook” division of classical dynamicalsystemsinto conservativeand dissipativesystems.We concludeby listing variousproblemsthat we considerinterestingfor thefuture, someof which are alreadybeingexplored:

(i) The local reversibility testsof sections3.3 and 3.4 allow us to identify somearea-preservingmappingslike (3.77)that arenot reversible.Wehavefollowed perioddoublingin suchone-parameter,nonreversible,area-preservingmappings. We find the parametersequenceof successiveperioddoublings is characterisedby the exponent~= 8.721... Togetherwith the resultsof chapter5, thissuggests6AP = ôAp,R = = 8.721..., where the subscriptsstand, respectively,for area-preserving,nonreversiblemappings; area-preserving,reversible mappings;and non area-preserving,reversiblemappings(symmetricperiod doubling in the last case).The number8 = 8.721...thereforecharacter-ises a large perioddoubling universalityclassamongmappingsof the plane.

It would neverthelessbe interestingto seeif and how the lack of reversibility of area-preservingmappingslike (3.77)could be detectedin thegeometryof thephaseplaneof themapping.This maylead to an independentgeometrictest or criterion for identifying reversiblemappings,which concen-tratesmoreon their global reversibility thanon theirlocal reversibility. As pointedout in section3.4,the following is still an openquestion:is Rannou’smapping,

164 J.A. G. Robertsand G.R.W. Quispel,Chaosand time-reversalsymmetry

1.3x102

~ ~.

—,

1 3x102 1 3x10—S

- a: -.

‘Is’..

1.3x10—2

1.3x106

(b)~i..

—1 3x10—6 _____________________ X 1 3x10—6

1.3x10—6

Fig. 8.1. Phaseportraitsof example2 of table4.1 with C = 0.3 and: (a) a= 102;(b) a = 106. Theorigin is anelliptic fixed pointwhenC = 0.3 for alla. The asymmetricfixed points are at (1/a, —1/a) and(—1/a, 1/a).

R:x’=x+y+l—cosy, y’=y—Csinx’—C(l—cosx’)

a reversiblemapping of the plane?(ii) It would be pleasingto extendthe study of reversibility to higher-dimensionalmappings.For


example,Feingoldet al. [1988]give the following exampleof a three-dimensionalvolume-preservingmapping derivedfrom the ABC flow in fluid dynamics,

x’ =x+ Asinz+ Ccosy,LABC: y’=y+Bsinx1+Acosz,

z’ = z + C sin y’ + B cosx’,

which is reversiblein the caseA = B (seealso Turnerand Quispel [1992]).(iii) Phaseportraitslike fig. 1.3 and figs. 4. lb or6.2 makeonewonderabouttheboundaryin phase

spacebetweentheconservativeanddissipative(and expansive)regionsin reversiblemappings.In figs.4.lb or 6.2 this boundaryis providedby the innermostKAM curve, whereasin fig. 1.3 it is less clearwhereone regionendsandtheotherbegins. In fig. 8. la,b we showan attemptto bring theattractingasymmetricfixed point (lIe, —1/s) in example2 closeto theKAM curvesaroundthe origin by takings—~ [from appendixB when~ <C < ~ and s is positive, the fixed point (1Is, —1 Is) is an attractingfixed point and, moreover,the origin is stable].We find that the KAM region contractsaroundtheorigin in proportionto theproximity of theattractor,producinganappealingself-similarity. It would beinterestingto investigatethe propertiesof theboundarybetweenthe two features.

(iv) In broaderterms, the similarities betweenthe dynamics in Hamiltonian systems and thedynamicsassociatedwith symmetric periodic orbits of reversiblesystemsleadsto asking if all theimportant “conservative” theoremsand results go through to reversible systemsunder suitableconditions;e.g., at the level of mappings,is therea reversibleAubry—Mathertheoremto mirror thearea-preservingtheorem (cf. Arrowsmithand Place [19901for discussionof the area-preservingcase).Some thoughts about the possible constructionof a unified theory of Hamiltonian and reversibledynamicalsystemsare given by Sevryuk[1986,p. 293], who also lists somefurther extensionsandgeneralisationsthat could be madeto the theory of reversible systems (see also conjecturesanddiscussionin Sevryuk [1991a]).These include relaxing the analyticity assumptionsin the reversibleKAM theorem(Sevryuk[1991a]statesthat the C~’and finite differentiability versionsare expectedtohold), developingan infinite-dimensionalreversibleKAM theory and studying Arnol’d diffusion inreversiblesystems.Other possibilitiesinclude the study of reversiblesystemson othermanifolds andtopological (i.e. nondifferentiable)reversible systems. It appearsthat reversiblepartial differentialequationsareimportantfor physicalapplicationsandso shouldalso bewell treatedfrom the theoreticalpoint of view.

Acknowledgements

This reportis basedupon thePh.D. thesisof the’first authoratMelbourneUniversity,Australia(cf.Roberts[1990a]).We are grateful to Cohn Thompsonand Hans Capelfor their supportand manyuseful suggestions.We havegreatly benefittedfrom readingthe significant contributions of Greene,MacKay andcoworkersto the study of reversible,conservativesystems.We thankM.B. Sevryukforongoing correspondence,including a useful bibliographyon reversiblesystemscompiledby him (cf.Sevryuk [1991b]).J.A.G.R. is grateful to the Stichting voor FundamenteelOnderzoekder Materie(FOM) for financial support.


Appendix A. Integrablereversiblemappings

In this appendixwe give someexamplesfrom a specialclassof reversiblemappingsof the plane,namely the integrablereversiblemappings.The twist mapping (5.1) is the canonicalexampleof anintegrablemapping,with integralI = r, the radius. By analogy,our working definition of “integrable”for a moregeneralmappingof theplaneL: (x, y)—~(x’, y’) is that themappingbe measure-preserving(cf. section2.2) and possessan integralI(x, y) definedby

I(x’, y’) = I(x, y) (A.1)

for all (x, y) in the mapping’sdomain. I is evidently a constantof the (mapping’s)motion. It follows

that the orbits of all points in theplanelie on the level curvesgiven byI(x,y)=C, (A.2)

whereC for any orbit is determinedfrom the initial condition,i.e., C= I(x0, y0). Thereareno chaoticorbits. Anotherway of expressingthis is that thephasespaceof anintegrablemapping(i.e., theplane)is foliated by invariantcurves(A.2) [cf. (2.6) and (2.7) for the definition of an invariantcurve], andthat the motion on any invariant curve is regular,i.e., periodicor quasiperiodic,asin the canonicalexampleof the integrabletwist mapping given in (5.1). This working definition then mirrors thepropertiesof an integrableHamiltonian system with n degreesof freedom (cf. Whittaker [1927],Goldstein [1950],Percival [1986],Arnol’d [1988]),which hasn independentconstantsof motionthatare in involution, and linear motion on the n-dimensionallevel set defined by these constants.Commonly, the level set is equivalentto an n-torusand themotion is periodicor quasiperiodicon thetorus.

One of the first people to study integrablemappingswas McMillan [1971].He introduced themapping

L1:x’=y, y’=—x— PY+EY (A.3)ay +/3y+y

where a, f3, y and s are arbitrary constants.This mapping is area-preservingand is reversiblewithinvolution x’ = y, y’ = x (cf. table3.1). It is also integrablebecauseit preservesthe following integral:

11(x, y) ax2y2 + f3(x2y+ xy2)+ ‘y(x2 + y2)+ sxy. (A.4)

It follows that theone-parameterfamily of symmetricbiquadraticcurvesI~(x, y) = C, parametrisedbyC, foliates the phase plane of (A.3). Each curve in this family is symmetric about y = x anddouble-valued.

A more generalarea-preservingand reversibleintegrablemappingis given by*)

L2:x’=—x— 6y

2+ey+~ , ~x’2+ex’+A (A.5)ay2-i-.8y+y ax’ +15X’+K

*) This mappingcan alsobe quantized,while retainingits integrability [Quispeland Nijhoff 1992].

l.A.G. Robertsand G.R.W.Quispel,Chaosand time-reversalsymmetry 167

The map L2 can be written as the compositionof two involutions, i.e., L2 = H2 ° G2 with

— 6y2+sy+~

H2: , f3x

2+sx+A G2: x — X ay

2+f3y+y’ (A.6)Y Y 2 ‘

ax +6x+K )‘ —)‘.

The mapping L2 hasthe following integral:

12(x, y) = ax2y2 + /3x2y+ yx2 + 8xy2 + sxy + ~x+ ~ + Ay. (A.7)

Eachof thecomponentinvolutions of L2 also preservestheintegral (A.7). The one-parameterfamily of

biquadraticcurves12(x, y) = C that foliatesthe planeare now not in generalsymmetricabouty = x.This is illustrated in fig. A. 1 which is a phaseportrait of a typical mappingof the form (A.5).

Finally, the most general integrablereversible mapping of the plane found to date is given byQuispel et al. [1988,1989],

L .X,f1)xf2~) ,g1(x’)—yg2(f) (A8)3. f2(y)—xf3(y) ‘ )‘ g2(x’)—yg3(x’) ‘

where the functionsf1 and g1 arein generalquarticpolynomialsdefinedby

1(x) = (f1(x), f2(x), f3(x)) = (A0X)x (A1X), (A.9a)

g(x) = (g1(x), g2(x), g3(x))= (A~X)x (A~X), (A.9b)

-15 15

Fig. A.!. Typical invariant curves of the asymmetric integrable mapping (AS) with a = y= a = 1, fi = 6 0, a = —2, A = —0.2, ~= —8. Thismappinghasan elliptic fixed point atabout(4.01,0.24).All of its invariantcurvesareclosed.A representativeinitial pointP, on acurvewith I = 1,is takento the intermediatepoint T by the involution G2. The point T is then taken by “2 to the point P,, which is the imageof P,underthemapping(A.5) (L2 = H2 G2). For typical 1~,successiveiterates of P, fill out the entire curve I = I~.

168 l.A.G. Robertsand G.R.W. Quispel, Chaosand time-reversalsymmetry

x2\ y2

X:= Y:= y , (A.10)a~ /3. ~

A~:= 6~ ~, ~ , (A.11)K, A. ,jL,

where a., 13,, y,, 6,, e,, ~, ,c~,A,, ii., (i = 0, 1) are 18 arbitrary parameters.The mapping (A.8) isreversiblesinceit can be written as thecompositionof two involutions, i.e., L

3 = H3 ° G3 with

= x,

H3: , — g1(x) —yg2(x)

— ~2(x)— yg3(x) ‘ (A.12)

,f1(y)—xf2(y)G3: f2(y)-xf3(y)’

y,—y.

The mapping L3 hasthe following integral:

I (x ) = a0xy + /30X y + y0x2 + 8

0xy2 + s

0xy + ~x + ~0y2+ A

0y +~ y a1x

2y2 + /31x

2y + y1x

2 + 81xy

2 + s1xy + ~1x+ ic~y

2+ A1y +

=(X.A0Y)I(X.A1Y).

The mappingL3 is not area-preserving,but it is measure-preserving(cf. section2.2) with a density

m3(x, y) = (a1x2y2 + f3

1x2y+ -y

1x2 + 6

1xy2 + s

1xy + ~1x+ K1y2+ A

1y+

A way of proving this measurepreservationis to utilize the decompositionof L3 as a productofinvolutions, rememberingthatany involution is measure-preserving[cf. eq. (3.6)]. Although weknowthat thecompositionof two measure-preservingmappingsis notnecessarilymeasure-preserving,wecanusethe fact that the involutionsH3 and G3 also havethe integral13(x, y) to showthat they sharethesamedensity, which guaranteesthat their composition L3 is measure-preservingwith this densityaswell. In this regard,it is easyto provethat if ameasure-preservingmappingwith densitym(x, y) is alsointegrablewith invariant I(x, y), thenany function of I(x, y) multiplied by m(x, y) is anotherdensityfor the mapping. Using this property,a commondensity for both of thecomponentinvolutions of L3canbe found[Roberts1990a, 1992]. It shouldbe notedabovethat the mappingL2 is aspecialcaseofthe mappingL3, and that in turn the mappingL1 is (for all intentsandpurposes)a specialcaseof L2.

Up to now we havenot commentedon the motion on the invariant biquadraticcurvesof theseintegrablemappings.On the closedbiquadratics,it is in fact quasiperiodic,or elseperiodicwith thecurve comprising an infinity of adjacentn-cycles. On unboundedbiquadratics,the motion typicallydivergeslinearly to infinity. The dynamicson the biquadraticscan be deducedfrom the fact that abiquadraticcurve can be parametrisedin terms of Jacobianelliptic functions [Roberts1990a1. Veryrecently,measure-preserving,reversibleintegrablemappingsof r havebeendiscovered(cf. Papageor-

l.A.G. Robertsand~G.R.W. Quispel, Chaosand time-reversalsymmetry 169

giou et al. [1990],Quispel et al. [1991];seealso Capelet al. [1991]and referencestherein for otherrecentdevelopmentsin the field of integrablemappings;also cf. Bellon et al. [1991a,b, c] for reversiblemappings that possessalgebraicintegrals and arise from integrablestatistical-mechanicalmodels).(Note that the definition of integrability of mappingsof W is still underdiscussion.)

Appendix B. Nonlinearstability analysisfor elliptic fixed points

In this appendix,wegive a quantity that canbecalculatedfrom theexpansionto third orderaroundanelliptic fixed point andhelpsto decidethe fixed point’s stability. An exampleoftheevaluationofthisquantity is providedfor example2 of table4.1. When a fixed pointx0 of a mappingis elliptic, it hasaJacobiandeterminantJ(x0) = 1 and theeigenvaluesof theJacobianevaluatedat x0 aree~’°with 0 ~ 0,IT. In this casethepoint is linearly stable.The actualstabilityof thepoint cansometimesbedeterminedby studying the higher order nonlinear terms of the expansionaroundthe fixed point. Quantitiesinvolving successivelyhigher degreecoefficients of this expansioncan be calculatedto determinewhetherthe point is stableor not (ci. Bernussou[1977,chapter11.4.1] andMira [1987,chapter5.7]).Herewe give the first suchquantity, denotedby G1, which involves coefficientsof secondand thirddegreetermsof the expansion.

Assumethat the mapping aroundthe fixed point x0 hasbeentakento the form

x’ =&°x+f20x2+f

11xy+f02y2+f

30x3+f

21x2y+f

12xy2+f

03y3 +...,

(B.1)y’ =e’10y+f~x2+f~

1xy+f~y2+f~x3+f~x2y+f~

1xy2+f~y3+...,

where * denotescomplex-conjugate(the way to obtain this complexform of the mappingexpansionfrom the real form is describedat the end of section2.2). Thendefine

3i0 2 ‘° 2 ~ 3b0

~ l_~b0)~2Jh1(l~eb8+

(B.2)

This quantity is obviously real. Thenprovided0 ~ 2kirI3, k = 1, 2, and 0 ~ 2kirI4, k = 1, 3, (aswell asthe original assumption0 ~ 0, ir) we have(cf. Mira [1987,proposition5.7])

(i) if G1 <0, the fixed point x0 is spirally attracting,

(ii) if G1 >0, the fixed point x0 is spirally repelling,(iii) if G1 = 0, thestability of the fixed point is undetermined.

In the last case,(iii), one may try and useother quantitiesG,, i> 1, which involve higher degreecoefficients(and so becomevery complicated)to determinestability in a similar way. If we write outthe third order condition for the mapping expansion(B.1) to be measure-preserving(via table 2.1),which is equivalentto the third ordercondition that it betheexpansionof areversiblemappingarounda symmetricfixed point (via table3.2),we find that thecondition is equivalentto G1 = 0. This is to beexpectedbecausea fixed point of a measure-preservingmapping, or a symmetric fixed point of areversiblemapping,cannotbe attractingor repelling— thecaseG1 ~ 0.

We usethe quantity G1 to investigatethe stability of the fixed point (lIs, —1Is) of the mappingexample2 of table 4.1,


x’ ~2~2~3 , y’—x+2C~+2~2 (~y+s2x2y+2e2x3). (B.3)l+sy

This point hasJ = 1 for all Candall s and,for any s, is elliptic whenC is in the range~ <C < ~. Whenit is elliptic, it is stablein the linear approximation.We havenumerically studiedthis examplewhens= 1.1 and found (its, —lIe) = (0.9090... ,—0.9090...) to be spirally attractingwhenit is elliptic(when s =20 we find the samebehaviour,as shownin fig. 1.3).

Insteadof using (B.3) in theanalysis,considerthemappingobtainedfrom it via the transformationx—+x/s,y--~y/s

x’ = — ~ , y’ = —x + 2C~+ 2M2Is (~ = y + x2y + 2x3). (B.4)1+y

In this form of themapping,the fixed point(lIs, —1 Is) is takento (1, —1) for all e andC. Its stabilityis unaffectedbecausestability is a conjugacy-invariant.The expansionof (B.4) aroundthefixed point(1, —1) to third order is

x’=4(1—4C)x+(1—8C)y+F20x

2+F11xy+F02y

2+F30x

3+F21x

2y+F12xy

2+F03y

3,(B.5)

y’ = (8C — 1)x+ 4Cy + G20x

2+ G11xy + G02y

2+ G30x

3 + G21x

2y+ G12xy

2+ G03y

3,

where the secondordercoefficientsare

F20=32C

2—12C—641s+1, G20’2(5C+16Is),

F11 = 4(8C2+ C — 161r), G

11 = 4(C + 81�),

F02 4(2C2+C—4Is), G

02—8/s,

and thethird ordercoefficientsare

F30 = 256C3+ 48C2 + 18C+ 256C1s— 128/s — 1, G

30 = 4(C + 20Is),

F21 = 2(192C3+ 36C2 + 9C + 192C/s— 401�), G

21 = 2(C + 36Is),

F12 = 4(48C3+ 10C2 + 48CIs + C + 21�), G

12 16/�,

F03=8(4C3+C2+4CIs+1Is), G

03~0.

From the linear partof this expansionit is seenthat J(l, —1) = 1 andTr(1, —1) = 4— l2C so that(1, —1) is elliptic for —2<4— 12C’<2, which gives the range~<C< ~quotedabove.With C in thisrange,apply to (B.5) the linear transformation

X=(A”1—A)x—By, Y=(A—A)x—By

A=4(1—4C), B=(1—8C), A=e’°: A2—(4—12C)A+1=0.

JAG. Robertsand G.R.W. Quispel, Chaosandtime-reversalsymmetry 171

This brings it into the complexform (B.l). CalculatingG1 we find

G1 = 2[e(2C — 1)(8C— l)(6C — 1)2]1 . (B.6)

SinceG1 <0 when~ <C < ~ands is positive, the fixed point (lIe, —1 Is) is anattractingfixed point in

example2.

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